1=0.999... infinities and box of chocolates..Phliosophy of Math...

[rpenner]

As always, $$T_{n} = 9 \times 10^{-n}$$.

So, $$\lim_{n \to \infty} \frac{T_{n+1}}{T_n} = \frac{1}{10}$$.

Correct.

Considering the limit theorems [Ref Theorem 400 in association with Remark 401 for item 3. in page 151], we can also write,

Your reference is correct. But you have not learned from it. Also it concerns limits of functions of real variables which is a subtly different topic than limits of sequences which is the topic here. You didn't understand examples 403 and 404 which follows from remark 402.

The subject you want to understand in relation to $$T_n$$ and $$\lim_{n\to\infty} T_n$$ and $$\lim_{n\to\infty} \frac{T_{n+1}}{T_n}$$ is covered in

• http://science.kennesaw.edu/~plaval/math4381/seq_limdef.pdf (Especially to understand example 300)
• http://science.kennesaw.edu/~plaval/math4381/seq_limthm.pdf ( Especially Remarks 316 and 320, Theorem 330 )

$$\lim_{n \to \infty} \frac {T_{n+1}}{T_n} = \frac {\lim_{n \to \infty} T_{n+1}}{\lim_{n \to \infty}T_n} = \frac{T_{\infty +1}}{T_\infty}=\frac{1}{10}$$

• $$\lim_{n \to \infty} \frac {T_{n+1}}{T_n} = \frac {\lim_{n \to \infty} T_{n+1}}{\lim_{n \to \infty}T_n} $$ is not true if $$\lim_{n \to \infty}T_n = 0$$ or if $$\lim_{n \to \infty}T_n$$ does not exist in the reals.
• $$\frac {\lim_{n \to \infty} T_{n+1}}{\lim_{n \to \infty}T_n} = \frac{T_{\infty +1}}{T_\infty}$$ is not true if $$T_{\infty+1} \neq T_{\infty}$$ or if $$\lim_{n \to \infty}T_n$$ does not exist in the reals or $$\lim_{n \to \infty}T_n$$ is zero. Additionally, it does not follow unless $$T_{\infty} = \lim_{n \to \infty}T_n$$ which does not follow from the laws of arithmetic of reals.
• And the statement $$\frac{T_{\infty +1}}{T_\infty}=\frac{1}{10}$$ does not follow from arithmetic in the reals (which is the only type of arithmetic your cited source supports. Additionally, it does not follow in the extended reals. Additionally, it is an incorrect statement in the hyperreals, which you would understand if you ever did arithmetic in the hyperreals.

So you have combined three untrue statements in a row. Now it is true that $$\lim_{n \to \infty} \frac {T_{n+1}}{T_n} = \frac{1}{10}$$ but you did the opposite of demonstrating that.

As the number $$\frac{1}{10}$$ is not undefined, from the above we can say that $$\lim_{n \to \infty}T_n \ne 0$$.

Incorrect and baseless. The only way you can say this is to ignore the definition of the limit of a sequence. (Definition 287 from your reference.)

 

[hansda]

As always, $$T_{n} = 9 \times 10^{-n}$$.

Let us assume $$\lim_{n \to \infty} T_n = 0$$.

If we follow the definition of a limit of a sequence, we can not ignore the $$\epsilon$$ factor.

So, here $$T_n < \epsilon $$, for any $$n > N $$. [ Consider $$n, N, \epsilon$$ as defined in the link].

Also $$T_n \ne 0$$ because $$\lim_{n\to\infty} \frac{T_{n+1}}{T_n} = \frac{1}{10}$$.

So, we can say $$0<T_n<\epsilon$$ for all $$n > N$$.

 

[rpenner]

Let us assume $$\lim_{n \to \infty} T_n = 0$$.

You don't assume what you want to prove is true. (That would be a circular argument.) You state what you hold to be true and use axioms, theorems and definitions to prove the result from the givens. Alternatively, you assume the opposite of what you want to prove and demonstrate from the axioms, theorems and definitions that a contradiction results from assuming the opposite of what you want to prove is true.

If we follow the definition of a limit of a sequence, we can not ignore the $$\epsilon$$ factor.

You are the one who didn't understand this.

Formal Definition
We call $$x$$ the limit of the sequence $$(x_n)$$ if the following condition holds:

For each real number $$\epsilon > 0$$, there exists a natural number $$N$$ such that, for every natural number $$n > N$$, we have $$|x_n - x| < \epsilon$$.​

In other words, for every measure of closeness $$\epsilon$$, the sequence's terms are eventually that close to the limit. The sequence $$(x_n)$$ is said to converge to or tend to the limit $$x$$, written $$x_n \to x$$ or $$\lim_{n \to \infty} x_n = x$$.

If a sequence converges to some limit, then it is convergent; otherwise it is divergent.

http://en.wikipedia.org/wiki/Limit_of_a_sequence#Formal_Definition​

But why are you changing authorities just because your last attempt to invoke authority backfired so badly? The authorities you cite are both summarizing the same body of standard math (real analysis) and therefore both give equivalent definitions.

Definition 285 (Eventually and frequently) Let $$(x_n)$$ be a sequence and $$ S \subseteq \mathbb{R}$$.

1. $$(x_n)$$ is eventually in $$S$$ if there exists $$N \gt 0$$ such that $$n \geq N \Rightarrow x_n \in S$$.
2. $$(x_n)$$ is frequently in $$S$$ if $$x_n \in S$$ for infinitely many values of $$n$$.

Definition 287 (Limit) $$\lim_{n\to\infty} x_n = L$$ if for every neighborhood $$U$$ of $$L$$, $$(x_n)$$ is eventually in $$U$$. In this case, we write one of the following:

$$ \begin{array}{c} \lim_{n\to\infty} x_n = L \\ \lim x_n = L \\ x_n \to L \; \textrm{as} \; n\to\infty \\ x_n \to L \end{array}$$​

Theorem 289 (Finite Limit) Let $$( x_n )$$ be a sequence, and $$L \in \mathbb{R}$$. The following are equivalent:

1. $$L$$ is a limit of $$( x_n )$$ in other words $$\lim_{n\to\infty} x_n = L$$.
2. $$\forall \epsilon \gt 0$$, $$( x_n )$$ is eventually in $$ \left( L - \epsilon, \quad L + \epsilon \right)$$.
3. $$\forall \epsilon \gt 0 \exists N \gt 0$$, such that $$n \geq N \Rightarrow x_n \in \left( L - \epsilon, \quad L + \epsilon \right)$$.
4. $$\forall \epsilon \gt 0 \exists N \gt 0$$, such that $$n \geq N \Rightarrow | x_n - L | \lt \epsilon$$.

http://science.kennesaw.edu/~plaval/math4381/seq_limdef.pdf​

This is the same idea as in Wikipedia rephrased in just slightly different language and understanding this is a prerequisite making a claim that either authority supports one's case.

So, here $$T_n < \epsilon $$, for any $$n > N $$. [ Consider $$n, N, \epsilon$$ as defined in the link].

You are citing the definition but you are neither using the definition or even demonstrating understanding of the definition. If there is a certain relation between $$N$$ and any legal value of $$\epsilon$$ that ensures that if $$n \gt N$$ then $$|x_n - x| < \epsilon$$ that proves $$\lim_{n\to\infty} x_n = x$$. In symbolic form we can write:

$$\forall \epsilon \in \mathbb{R} \left( \epsilon \gt 0 \, \Rightarrow \, \exists N \in \mathbb{N} \forall n \in \mathbb{N} \left( n \gt N \; \Rightarrow \; | x_n - x | \lt \epsilon \right) \right) \quad \Rightarrow \quad \lim_{n \to \infty} x_n = x$$
"IF it is the case for every epsilon in the Real Numbers, if epsilon is positive then there exists a specific natural number N such that for every natural number n the fact that n is larger than N means x_n is closer to x than epsilon THEN the limit of the infinite sequence (x_n) is x."​

So this definition requires finding a relation between the infinite sequence (x_n), all positive numbers epsilon, some function N and the unique real number x which does not depend on epsilon and N. This is easy to do for a decreasing geometric series like (T_n).

Specifically for all positive real numbers $$\epsilon$$ the following holds $$N \gt f(\epsilon) = - 1 - \log_{10} \, \frac{\epsilon}{9} $$. If you want to make it concrete, you could use the floor function and the maximum function to write $$N_{\textrm{min}}(\epsilon) = \textrm{max} \left( 0, \quad \left\lfloor - \log_{10} \, \frac{\epsilon}{9} \right\rfloor \right)$$
So $$N, n \in \mathbb{N}$$ and $$N \gt f(\epsilon)$$ and $$n \gt N$$ means $$ n \gt - \log_{10} \, \frac{\epsilon}{9}$$ and thus $$-n \lt \log_{10} \, \frac{\epsilon}{9}$$ and $$10^{-n} \lt \frac{\epsilon}{9}$$ and $$| T_n - 0 | = T_n = 9 \times 10^{-n} \lt \epsilon$$ for any positive real number epsilon. Thus this proves $$\lim_{n\to\infty} T_n = 0$$.
Also, $$N, n \in \mathbb{N}$$ and $$N \geq N_{\textrm{min}}(\epsilon)$$ and $$n \gt N$$ means $$n \gt - \log_{10} \, \frac{\epsilon}{9}$$ with the same result of proving $$\lim_{n\to\infty} T_n = 0$$.

Also $$T_n \ne 0$$ because $$\lim_{n\to\infty} \frac{T_{n+1}}{T_n} = \frac{1}{10}$$.

No axiom, theorem or definition connects those two assertions. Thus it is a non sequitur error in your argument. The first is, as written, as statement about a specific, unspecified term in an infinite sequence. The second is a description of the collective behavior of nearly all the elements of the sequence. Indeed, one may even write: $$\forall n \in \mathbb{N} \left( \frac{T_{n+1}}{T_n} = \frac{1}{10} \right)$$ and be exactly correct. Now this statement implies $$\forall n \in \mathbb{N} T_n \neq 0$$ which follows by the rules of arithmetic. But neither of these is a statement in analysis, and neither contradicts $$\lim_{n\to\infty} T_n = 0$$.

So, we can say $$0<T_n<\epsilon$$ for all $$n > N$$.

For the reasons above, you have got the notation wrong and and required to specify N before you can assert that.
$$\forall \epsilon \in \mathbb{R}^+ \forall N \in \mathbb{N} \forall n \in \mathbb{N} \left( N \geq \textrm{max} \left( 0, \quad \left\lfloor - \log_{10} \, \frac{\epsilon}{9} \right\rfloor \right) \quad \wedge \quad n \gt N \quad \Rightarrow \quad 0 \lt T_n \lt \epsilon \right)$$
And from this , the following is true:
$$\forall \epsilon \in \mathbb{R} \left( \epsilon \gt 0 \Rightarrow \forall N \in \mathbb{N} \forall n \in \mathbb{N} \left( N \geq \textrm{max} \left( 0, \quad \left\lfloor - \log_{10} \, \frac{\epsilon}{9} \right\rfloor \right) \quad \wedge \quad n \gt N \Rightarrow | T_n - 0 | \lt \epsilon \right) \right)$$
And from this, the following is true:
$$\forall \epsilon \in \mathbb{R} \left( \epsilon \gt 0 \Rightarrow \exists N \in \mathbb{N} \forall n \in \mathbb{N} \left( n \gt N \Rightarrow | T_n - 0 | \lt \epsilon \right) \right)$$
And from this, it follows that
$$\lim_{n \to \infty} T_n = 0$$

 

[Zman Kietilipooskie]

does 1=.9999...., no because .999.... always is infinitely .00....1/1 less then 1 is (.... being the same amount for both situations) so 1 does not equal .999..../1. So if you take for example (being that it is a infinitely repeating pattern of 1-....1) 1-.9999 you get .0001 and if that is not true then and infinite example would not apply as well. Also .999... means that you get as close to 1 as possible but never actually reach it. This is similar to concept in algebra of ()vs[] so for example (1) is not the same as [1].

 

[rpenner]

Introducing $$D_n = \frac{1}{9} T_n = 10^{-n}$$.

does 1=.9999...., no because .999.... always is infinitely .00....1/1 less then 1 is (.... being the same amount for both situations).

That is circular reasoning. You "explain" $$1 \neq 0.999... $$ "because" $$1 \gt 0.999... $$ which is a naked assertion, not an explanation.

So if you take for example 1-.9999 you get .0001 and if that is not true then and infinite example would not apply as well.

You are asserting that because $$1 - S_n = D_n$$ for all natural numbers, $$n$$ and then leap to the conclusion that this holds also in the case of an infinite number of digits. If you defined $$S = \lim_{n\to\infty} S_n, \quad D = \lim_{n\to\infty} D_n$$ then it follows that $$1 - S_n = D_n$$ for all natural numbers $$n$$ implies $$1 - S = D$$, but this does not demonstrate that $$S \neq 1$$ or $$D \neq 0$$.

Also .999... means that you get as close to 1 as possible but never actually reach it.

Except that statement about what $$0.999...$$ is defined in terms of real numbers and with real numbers it is impossible for two distinct numbers to be "as close to [each other] as possible but never actually reach [identity]."

Say x and z are two distinct numbers and you say $$x \gt z$$ AND there is no number $$y$$ such that $$x \gt y \gt z$$ is true. I say you are wrong. $$\epsilon = x - z$$ is obviously a number by the rules of real arithmetic. It follows that $$\epsilon \gt 0$$. $$\frac{1}{2}$$ is obviously a number by the rules of real arithmetic. Then it follows that $$1 \gt \frac{1}{2} \gt 0$$. Likewise $$\frac{1}{2} \times \epsilon \equiv \frac{\epsilon}{2}$$ is obviously a number by the rules of real arithmetic. It follows that $$\frac{\epsilon}{2} \gt 0$$. Then it follows that $$\epsilon \gt \frac{\epsilon}{2} \gt 0$$. And if we add z to all three numbers, it follows that $$z + \epsilon \gt z + \frac{\epsilon}{2} \gt z$$. Or $$x \gt z + \frac{\epsilon}{2} \gt z$$. Now from the rules of real arithmetic, it follows that $$ z + \frac{\epsilon}{2}$$ is a real number and if we call that $$y$$ then we have $$y = z + \frac{\epsilon}{2} = z + \frac{x - z}{2} = \frac{x + 2 z - z}{2} = \frac{x + z}{2} $$ which is between x and z, contrary to your position.

Likewise, if you say x and z are two distinct numbers and you say $$x \gt z$$ AND there is no number $$y$$ such that $$x \gt y \gt z$$ is true, I say you have a second contradiction. Let $$\epsilon = x - z \gt 0$$ as before. Then $$\epsilon$$ must be the smallest possible positive number. That means $$ 0 \lt \epsilon \lt 1$$, right? But that means that $$\frac{1}{\epsilon} \gt 1$$. By the Archimedean property of real numbers, it follows that there is an integer, $$N$$ greater than it. Thus we have, $$N \gt \frac{1}{\epsilon} \gt 1$$ and so $$ 0 \lt \frac{1}{N} \lt \epsilon \lt 1$$. Therefore $$ x \gt z + \frac{1}{N} \gt z$$ and again your assumptions fail.

Further, if $$x \gt z$$ and $$\epsilon = x - z $$ and $$\exists N \in \mathbb{N}^+ \; N \gt \frac{1}{\epsilon}$$ then it follows that $$2 N x \geq \lfloor 2 N x \rfloor = \lfloor 2 N ( \epsilon + z ) \rfloor = \lfloor 2 + N z \rfloor = \lfloor 2 + 2 N z \rfloor = 2 + \lfloor 2 N z \rfloor \gt 1 + \lfloor 2 N z \rfloor \gt \lceil 2 N z \rceil \geq 2 N z$$ thus $$x \gt \frac{ 1 + \lfloor 2 N z \rfloor }{2 N} \gt z$$ and there is at least one rational number between any two distinct reals.

In symbolic form: $$ \forall x \in \mathbb{R} \, \forall z \in \mathbb{R} \, \left( x \gt z \; \Rightarrow \; x \gt \frac{x + z}{2} \gt z \right)$$ or more generically $$ \forall x \in \mathbb{R} \, \forall z \in \mathbb{R} \, \left( x \gt z \; \Rightarrow \; \exists y \in \mathbb{Q} \, x \gt y \gt z \right)$$ where $$\mathbb{Q} \subset \mathbb{R}$$

Application:

$$\pi \gt \sqrt{ \frac{40}{3} - \sqrt{12}}$$, $$\epsilon = \pi \gt \sqrt{ \frac{40}{3} - \sqrt{12}} \approx \frac{241643}{4073901538}$$, $$16860 > \frac{1}{\epsilon}$$, and so $$105935 \gt 2 \times 16860 \times \pi \gt 105934 \gt 105933 \gt 2 \times 16860 \times \sqrt{ \frac{40}{3} - \sqrt{12}} \gt 105932 $$. Thus $$\pi \gt \frac{105933}{33720} \gt \sqrt{ \frac{40}{3} - \sqrt{12}}$$.

Another Example:

$$ \frac{\sqrt{13 \times 233}}{5^2 \times 137 \times 283 \times 1733 \times 17669} \gt \frac{\sqrt{3 \times 5 \times 11 \times 23 \times 71 \times 89}}{131 \times 21773 \times 1500198347}$$,
$$\epsilon = \frac{4278960237809261 \sqrt{3029}-29679565916675 \sqrt{23980605}}{126997682432891295445269327175} \approx \frac{7037}{9912456690077761}$$, $$1408619680273 \gt \frac{1}{\epsilon}$$, $$ 6 \gt \frac{2 \times 1408619680273 \times \sqrt{13 \times 233}}{5^2 \times 137 \times 283 \times 1733 \times 17669} \gt 5 \gt 4 \gt \frac{2 \times 1408619680273 \times \sqrt{3 \times 5 \times 11 \times 23 \times 71 \times 89}}{131 \times 21773 \times 1500198347} \gt 3$$. Thus $$\frac{\sqrt{13 \times 233}}{5^2 \times 137 \times 283 \times 1733 \times 17669} \gt \frac{2}{1408619680273} \gt \frac{\sqrt{3 \times 5 \times 11 \times 23 \times 71 \times 89}}{131 \times 21773 \times 1500198347}$$

 

[hansda]

You don't assume what you want to prove is true. (That would be a circular argument.) You state what you hold to be true and use axioms, theorems and definitions to prove the result from the givens. Alternatively, you assume the opposite of what you want to prove and demonstrate from the axioms, theorems and definitions that a contradiction results from assuming the opposite of what you want to prove is true.

The point which i want to prove is that, $$T_n \ne 0$$for any value of $$n$$; where $$n $$and $$N$$ are natural numbers and $$n>N$$.

From the theorem of ratio test, we can see that the constant ratio of the infinite geometric series is maintained throughout the series. If any member of this series is zero(ie if $$T_n = 0$$ for any value of $$n$$), this constant ratio($$\lim_{n \to \infty} \frac{T_{n+1}}{T_n}$$) of the infinite geometric series can not be maintained.

If $$T_n = 0$$; $$\lim_{n \to \infty} \frac{T_{n+1}}{T_n} = \lim_{n \to \infty} \frac{T_{n+1}}{0} = undefined $$.

Hence $$T_n \ne 0$$ for any value of $$n$$.

 

[rpenner]

The point which i want to prove is that, $$T_n \ne 0$$for any value of $$n$$; where $$n $$and $$N$$ are natural numbers and $$n>N$$.

For what purpose to you wish to prove that? Do you just want practice writing mathematical proofs? If so then you need more practice. Your claim is not relevant to the discussion since it was already covered in post #915 (which predates your notation of $$T_n$$) and a later post. Allow me to remind you:

From [post=3164396]post #915[/post]:

And for every n greater than or equal to 1, we can prove $$S_{n+1} = S_n + \frac{9}{10^{n+1}}$$ and $$\frac{9}{10^{n+1}} \gt 0$$ so $$S_{n+1} \gt S_n$$.
So presumably $$S_1 \lt S_2 \lt S_3 \lt \dots \lt S_{n+5}\lt S_{n+6}\lt S_{n+7} \lt \dots \lt S$$ and indeed this is provable from the rules of inequalities.

As $$\frac{9}{10^{n+1}} = T_{n+1}$$ this is pretty clearly not only saying every member of the series $$(T_n)$$ is greater than zero, but the logic of the next line depends on understanding that.

From [post=3165524]post #990[/post]:

An[d] there is no value of n which makes $$T_n = 9 \times 10^{-n} = 0$$ true, because $$T_n$$ is only defined for the counting numbers.
There are an infinite number of members of that family, one for each of the counting numbers, but all are positive, which is to say, greater than zero.

What is correct to write is $$\lim_{n \to \infty} T_n = 0$$ which builds on the previous observation that "any positive number is more than some (or all!) members of the family $$T_n$$. " -- Specifically it is saying for any positive number, there is a finite number, m, so that every member of the family $$T_n$$ where $$n \gt m$$ is closer to 0 than that chosen positive number. And since no matter how large m is, most counting number are larger than m, this means no matter how much you "zoom in" on 0 with a microscope, there are still an infinite number of the family of $$T_n$$ close to it.

Limits and suprema are two topics of analysis that follow from the properties of real numbers which in turn follow from the basic geometrical ideas that Euclid wrote about.

All members of the series $$(T_n)$$ are positive which means none equals zero. But $$T$$ is not defined as a member of the family $$(T_n)$$; instead $$T = \lim_{n\to\infty} T_n$$ -- a notation you don't understand despite citing math texts and Wikipedia pages.

From the theorem of ratio test, we can see that the constant ratio of the infinite geometric series is maintained throughout the series.

The ratio test doesn't say that. The ratio test says that if the limit $$\lim_{n\to\infty} \left| \frac{T_{n+1}}{T_n} \right|$$ exists and that limit is less than 1 then the related infinite series $$\sum_{n\geq 1} T_n = \lim_{n\to\infty} S_n = S$$ exists in the real numbers.
The ratio test is an implication, not a logical equivalence, because $$\sum_{n\geq 1} \frac{1 + (-1)^n}{n^2} = \frac{\zeta(2)}{2} = \frac{\pi^2}{12}$$

If any member of this series is zero(ie if $$T_n = 0$$ for any value of $$n$$), this constant ratio($$\lim_{n \to \infty} \frac{T_{n+1}}{T_n}$$) of the infinite geometric series can not be maintained.

1. That does not follow, because if just one member of the series were zero, there would be an unending sequence members past that one member and thus the value of the limit of the sequence of ratios of terms would be unaffected.
2. The limit of a geometric series is actually a more fundamental concept than the limit test as the proof of the limit test depends on the proof of the convergences of a geometric series, not the other way around, as demonstrated in the proof of the limit test theorem by your own source:

See the Theorem 5.5 (Ratio test) on page # 16 of infinite series by KEITH CONRAD.

I did you the service of quoting this proof in my [post=3170118]#1079[/post]​

If $$T_n = 0$$;

I think you have lost the thread. Noone in this discussion is asserting that there is a counting number n such that $$T_n = 0$$.

$$\lim_{n \to \infty} \frac{T_{n+1}}{T_n} = \lim_{n \to \infty} \frac{T_{n+1}}{0} = undefined $$.

This does not follow from assuming there is a value of n that makes $$T_n = 0$$. Likewise it does not follow from part 3 of Theorem 315 from http://science.kennesaw.edu/~plaval/math4381/seq_limthm.pdf which says nothing about $$\lim_{n \to \infty} \frac{T_{n+1}}{T_n}$$ unless $$\lim_{n \to \infty} T_n$$ exists and is not zero.

Hence $$T_n \ne 0$$ for any value of $$n$$.

Again, you are not applying any mathematical principle here. The fact that no element of the unending sequence $$(T_n)$$ is equal to zero has no bearing on the truth of the statement $$\lim_{n\to\infty} T_n = 0$$, and what you needed to understand was highlighted for you from your own source on this page:

Correct. Your reference is correct. But you have not learned from it.
...
The subject you want to understand in relation to $$T_n$$ and $$\lim_{n\to\infty} T_n$$ and $$\lim_{n\to\infty} \frac{T_{n+1}}{T_n}$$ is covered in

• http://science.kennesaw.edu/~plaval/math4381/seq_limdef.pdf (Especially to understand example 300)
• http://science.kennesaw.edu/~plaval/math4381/seq_limthm.pdf ( Especially Remarks 316 and 320, Theorem 330 )

Example 300 demonstrates $$\lim_{n\to\infty} \frac{1}{n} = 0$$ even though it is obvious that zero is nowhere in the sequence.

When will you tire of misrepresenting authoritative sources of instruction? Why do you bother to cite them when you don't learn from them well enough to do the problems or otherwise use the definitions? When will you tire of advocating opinions in a field where you have no expertise?

 

[Undefined]

Hi James. Sorry for the tardy reply, but Trippy banned me for 14 days after I pointed out his transgressions against site rules and other members. I will be sending you a PM as soon as I get time to compile it (unless Trippy bans me permanently first to prevent it!). Anyhow, I am very busy, as you may be aware by now, so I must be brief in this present reply to yours above. If by any chance Trippy permabans me before I PM you, I'd just like to say it was nice knowing you; and that I have the impression you have been doing your level best with the most blunt and oh-too-human 'modding' tools/staff available to you under the circumstances. Anyhow, to your post...

Undefined:

You've posted a huge long post, yet still failed to answer the question I asked you. I remind you:

1/2 + 1/4 + 1/8 + ... = 1. True or false?

This is a mathematical statement. Either you think it is true, or you think it's false. There's no half-way house. So, which is it?

Let's look at some of what you wrote. I really don't want to wade through all of it.



I have not demanded anything of you. Here you are, making post after post in this thread, yet you seemingly don't have an actual point of view on the simple question above. All I see is waffle.

Also, didn't anybody ever tell you that using ALL CAPITALS on the internet is considered to be SHOUTING? Do you want to look like some kind of deranged nut? Because that's the kind of person who mostly uses random capitals on the internet.



I'm having trouble unpacking what you even mean by all that. All those CAPITALS for emphasis, and you still haven't communicated.



Very often indeed, I'm sure. What's wrong with that?



Not by me. Not at this stage.

I wouldn't have thought that answering a simple question like the one above, about your own personal beliefs, would be as hard as it apparently is for you. Maybe you should just leave this thread alone.



Does this mean that you're leaning towards saying the statement above is false, then? I really can't tell.



Forget my arguments. How do you respond to post #915, above? That has put the nail in the coffin in this thread, as far as I'm concerned. What's your analysis of that?



Nobody has asked you about reality and unreality. For my part, I simply want to know whether you agree with the truth of the above mathematical statement. It's a formal mathematical statement in a formal mathematical system. So, within that system, is it true or false?

And, while we're at it, is it true that, in the same system, 0.999... = 1?

One you've answered those questions, then maybe we can start getting all philosophical and start pondering whether this formal mathematics actually works in "real" world, whatever that is.



Ok. So help me.

Explain to me what is patently incomplete about my orthodoxy. Because it isn't patent to me at the present time.



Do you include post #915 in that?



You seem to think that a number is a process. Or do I have you wrong?

Also, I'm puzzled as to what you think is "real" about a number such as 7 or 1 or pi. I'd say all numbers are abstractions. We can talk about the number 3, but it's an abstraction until we start talking about 3 sheep or 3 dollars or 3 tons of salt.



It now sounds like you're saying you have no idea whether 0.999... = 1 or not. Is that correct?



I think you've misunderstood Goedel. Of course you can use arguments within a formal system to prove statements made within that system. If you couldn't all mathematics would be useless. Goedel's theorems are about the completeness and consistency of formal systems.



So, tell me how REALITY would work out whether 0.999... = 1 or not.



I don't think Motor Daddy has pointed out anything of the kind to me.

Maybe you can explain without all the CAPITALS and waffle.



I thought it nicely demonstrated the point that 1 thing can be divided equally into 3 parts, which was an issue Motor Daddy disputed at the time. Now, he appears to be running away from that silly claim. But perhaps you'd like to take up his baton?



Nobody's talking "real" yet. This thread is about the mathematical truth or falsity of 0.999... = 1, isn't it?



This sounds (a) like you think that 1/2 + 1/4 + 1/8 + ... doesn't equal 1, and (b) like you think a number is a process.

Do you want to address either of these points in more depth?

May I remind you and everyone that, as the "Philosophy of Maths' in the title implies, and as the location of this thread under the "Philosophy" section further stresses, this thread was opened for the express purpose of delving behind the maths system itself as it is currently constructed and practiced. That is why my responses to you here have been directed and based on the Philosophy and starting axioms etc origins' and 'adequacy' etc. So please do not think I am trying to evade your questions just because I am replying in the context of the philosophy/axioms and practice which led to the current expressions/arguments you and others keep 'challenging' me to answer in the current maths context rather than in the maths Philosophy-and-Axioms-questioning context which is what this thread is all about.

If you have read my replies to arfa et al in the context of philosophy/axioms, you should have noticed that I pointed out that 'point' is a philosophical notion only, and not a 'self-evident' fact. Hence the potential for GIGO from that starting 'point' axiom, irrespective of how 'rigorous' may be the following 'logics' based on such NON-rigorous and NON-maths starting axiom/point; which is why I pointed out the undefined, undetermined and singular/infinite are inescapable in such an 'iffy' maths/logic flowing from such an 'iffy' starting point/axiom. I already posted to Trippy where the like/like construction is fraught with peril from the get-go, else 0/0 would be as valid and meaningful as 9/9 IF "0" is a number on the extended or 'ring' number line. I have also given my observations on '0' in many contextual settings, including the obvious physical reality context, so I will not repeat them here either. Moreover, the way 'infinity' is being bandied about in the current (supposedly rigorously logical maths treatments) is obviously flawed also from the get-go, so GIGO is also inevitable, as all the 'infinities' which arise at present well demonstrate (by the way, I have also mentioned before that I have identified blatant logical flaws in all the currently accepted Hilbert's Hotel type treatments/assumptions of infinities. I have effectively demoloshed Hilbert's Hotel and all its like infinity 'arguments/treatments'; and the details will be published in my ToE book).



So, in that context and in keeping with the thrust of this tread, I will now briefly reply to some of your and others 'assumptions' and 'challenges' as the current maths poses/accepts them as so-called' proofs (as the current maths 'produces' but which do not answer as 'proofs' when the whole starting/background axiomatic philosophy aspects are brought into it to examine the validity or otherwise in the wider context, including the reality context of physics which maths purports to 'model'). I begin with your above reiterated question:

1/2 + 1/4 + 1/8 + ... = 1. True or false?

I was loathe to answer this before because the expression as constructed (according to current maths axioms/assumptions) does not have regard to the 'infinitesimal of last effectiveness step' which the logical extension of "..." will eventually run up against under any reasonable reality/logical context that is not flawed by the inadequacies of the current maths axiomatic start from the purely philosophical notion of 'point' which allows an infinite-points extension without any end (that is a purely philosophical assumption BUILT INTO that and 0.999... etc type expressions. The problem is that, as I pointed out before, there must be an ultimate step quantum for any series whether in reality or in logic, else it is a purely philosophical expression and not any sort of rigorously mathematical expression. Hence the GIGO potential which I have been addressing by pointing out (as this thread requires) where the starting axioms/assumptions and following arguments may be at fault and lead to such mathematical non-sequiturs which when examined are found to be purely philosophical expressions, without any real maths meaning or capability of treating them to arrive at any rigorously valid conclusions/claims about what that "..." means in any context, let alone in a maths context which is ultimately found to be based on purely philosophical starting premises which have no regard for either the logic or the reality of an ultimate step quantum of effectiveness in all contexts, be they maths constructs or physics realities.

Hence my answer, in the specific philosophy-of-maths context and thrust of this thread, can only be:

It depends on whether your 'expression' can 'handle' that ultimate smallest quantum step to transition from one 'state/frame' to the next state/frame via a 'transitional singularity state of SUPER-positioning' factors/values which cannot be properly 'divorced from' logics and physics by purely ad hoc and invalid philosophical/arbitrary 'fixes' as one goes along an already flawed-from-the-get-go' mathematical system that still outputs undefined, undeterminable and infinities etc.

Now IF we include the inescapable (logically and physically) ultimate step quantum of minimal effectiveness (INFINITESIMAL) into the axiomatic system/treatments, then such an expression as you posted above will be read as "..." indicating there will BE some transitional infinitesimal (currently not included in the maths axioms/construct as it stands at present) which EFFECTIVELY TERMINATES the series and transitions that series into an END STATE of "1" which is also the BEGINNING STATE for whatever comes next from there.

See where I am answering within the purview and thrust of this 'philosophy of maths' thread? Please then do not accuse me of avoiding, just because you are talking about the maths as is while I am talking about the philosophy/axioms from which the maths-as-is has been apparently ad-hocly and non-rigorously 'developed' to the point of it becoming meaninglessly infested with undetermined, undefined and infinities and obvious ill-logics and miss-assumptions being built into it, as I have already observed with supporting logical/reality (not philosophical like the mathematicians have been basically doing with their 'proofs' and expressions etc) based arguments more than once now here and elsewhere.


And speaking of infinities and the absurdities which start the maths of such, I draw your attention to this from rpenner to hansda:

It doesn't matter what I believe. It matters which mathematical axioms I choose to work with. Different axioms, different results. In this case $$\infty$$ as a quasi-number is only defined in one of the two systems I talk about and in that system $$T_{\infty+1} = \frac{1}{10} T_{\infty} = 0$$.

But neither of those two systems is necessary to everything in [post=3164396]Post #915[/post] being correct.

Now I ask you, James R, everyone, how can maths system 'choose' to start/base their following logics and treatments and results from such an obviously absurd 'axiom'?

Consider: Once an "$$\infty$$" has been invoked into an expression, how can anyone logically, physically (or even philosophically for that matter!) possibly pretend that they can "add 1" to such a thing?

The invocation of an infinity effectively creates a boundary condition beyond which there is only a DIFFERENT STATE and so no longer within that 'starting' infinity as invoked. In short: If we add 'anything more' to one infinity as invoked, it becomes ANOTHER infinity altogether. The former bounds its starting context; and any variation to it is impossible in any sense, and only can CHANGE the NATURE of that starting infinity to some other 'infinity' conceptualization.

See where the maths as currently formulated and 'practiced' wants it 'all ways' but cannot deliver in a 'consistent way, hence all the slippery a hoc assumptions/philosophical 'fixes' necessary to 'make it work at all'? The undefined, undetermined, and slippery logics/non-sequiturs (as perfectly exampled by that above axiomatic 'choice/treatment' in order to 'allow' that above absurdly dreamed up $$\infty+1$$ nonsense) should obviously be, especially in the absence of any real 'current maths' understanding of what infinities and infinitesimals really are, a cause of serious concern to all thinking people who can go beyond the circuitous and axiomatically inadequate self-selecting illogics/constructs currently being claimed as 'rigorous maths system'. The problem with that claim to 'rigour' etc, as I have shown already, is that it starts out all wrong, and has become a SELF-PERPETUATING GIGO system which needs on-the-fly 'fixes and non-sequiturs' to 'paper over the cracks which extend from the starting 'point' philosophy that makes nonsense of all the rest and prevented mathematicians from actually starting from axioms which are reality/logically founded and which would if followed properly provide understanding/treatments of infinities, infinitesimals and 'zero' and etc which void all the current undetermined/undefined etc nonsense outputs which perfectly demonstrate the inadequacy of the current maths/axioms/assumptions/treatments etc.

That above, and my previous here and elsewhere, is basically all I have to say on these matters until I publish the whole lot complete. Good luck and enjoy your further discussions/explorations, James, everyone!


James, I have been as brief as I could while still providing you with the essential thrust of my points/observations in the context of this thread regarding the philosophy etc of the maths and how I have been working to improve it so it can be started again from real/logical axioms instead of purely slippery/philosophical 'notions' it has been riddled with from the get go. As I 'pointed' out many times already. I have run out of time for now. So I will get back to you via PM about that 'other matter' (if I am not permabanned by 'you know who' before I get a chance to put my version/evidence to you!). Either way. No hard feelings, mate. I mean that sincerely, as always (regardless of whether certain troll-mods still have a hard time believing such a thing is possible given their individual/collective transgressions history).

Cheers James. Take care not to let your deference to certain 'types' lead your intellect and integrity into being too easily 'swayed' and 'captured' by the 'maths-without-sense' crowd and/or 'mod-troll gang leftovers' from the bad old days. Without true examination of 'orthodoxy' as currently assumed/practiced, no further real advance to towards the complete maths/physics ToE will be possible. Hence my taking risks such as these to point out the obvious flaws at present. I am entirely independent researcher/thinker upon the universal phenomena, and beholden to no 'group think' or 'publish or perish' imperative' or 'peer pressure grouping' etc crowd. I trust that will have been made abundantly clear by now to you and all thinking observers/forummers? Bye for now!

 

[arfa brane]

If you have read my replies to arfa et al in the context of philosophy/axioms, you should have noticed that I pointed out that 'point' is a philosophical notion only, and not a 'self-evident' fact.

Try this thought experiment: draw a pair of lines on the ground so they intersect somewhere.
Next, explain how this intersection is not a self-evident fact, but a mere philosophical notion.

You might have to skip over another self-evident fact: you can't draw lines on a real physical surface so they have only one dimension, so they won't intersect at a zero-dimensional point. But don't panic, philosophy can actually help with this by allowing one to abstract lines which are not physically one-dimensional so they are one-dimensional "abstractly".

 

[Undefined]

Try this thought experiment: draw a pair of lines on the ground so they intersect somewhere.
Next, explain how this intersection is not a self-evident fact, but a mere philosophical notion.

You might have to skip over another self-evident fact: you can't draw lines on a real physical surface so they have only one dimension, so they won't intersect at a zero-dimensional point. But don't panic, philosophy can actually help with this by allowing one to abstract lines which are not physically one-dimensional so they are one-dimensional "abstractly".

Hi arfa. I was just editing typos and saw this. Like I say, I haven't much time lately so briefly...

Your beginning assumption about 'lines' is flawed. All that follows is your own flawed results, not reality or logics. That is one of the current problems with all the inbuilt INSIDIOUS philosophical assumptions in maths which training and constant misunderstanding has inculcated in even the very best mathematical constructs/minds.

Actually, even your 'jesting words' tacitly acknowledge that 'mathematical lines' don't exist in reality physics. Even a infinitesimally 'slender' a 3-dimensional 'object' is composed of minimal physical 3-D quantum scale/dimensions of 'physical effectiveness'. We 'philosophically' run a 'abstract line' construct along its imagined physical 'centre of 3-D dimensional effectiveness' as a NOTIONAL 'axis line'.

Also, I have identified (as I alluded to in an earlier post) a 'proof' from geometry/topology which demolishes all the joint/disjoint arguments about 'philosophical' point, 'infinities of points', different infinities etc.

Sorry I can't say any more until I publish the lot. I have to leave it at the above salutary reminder of where the non-sequitur assumptions and philosophical 'points' and 'one-dimensional lines' NOTIONS ONLY have led you to make your non-sequiturs 'examples/arguments' above; even as you re-iterate them from 'textbook orthodoxy', you still don't see the inbuilt/inculcated non-sequiturs that those are based on from the get-go.

I hope you haven't take umbrage at my necessarily short and to the point observation there? I have to go. Bye!

PS: If I survive to visit another day, I will check in to see what transpires. Until then, arfa, good luck and thanks for the reasonable/polite discussion irrespective!

 

[arfa brane]

Your beginning assumption about 'lines' is flawed.

That's easy for you to say, but suppose I say just as glibly: "No, it isn't flawed". Are you really saying I can't draw lines on a surface so they intersect?

All that follows is your own flawed results, not reality or logics.

So what do all those groundsmen do who mark out lines on a football field? How are their results flawed?

Actually, 'mathematical lines' don't exist in reality physics.

In mathematics, physics doesn't exist. A physical cube has a distance between two diagonal vertices given by a mathematical formula; this formula also works with non-physical cubes.

 

[Undefined]

That's easy for you to say, but suppose I say just as glibly: "No, it isn't flawed". Are you really saying I can't draw lines on a surface so they intersect?

So what do all those groundsmen do who mark out lines on a football field? How are their results flawed?In mathematics, physics doesn't exist. A physical cube has a distance between two diagonal vertices given by a mathematical formula; this formula also works with non-physical cubes.

You are mixing frames of contexts. The concept of 'line' abstraction is philosophical overlay upon minimally slender objects in reality. The further use of the 'convenient abstraction' of a 'line' is inherent when you paint a line on the ground etc. The 'stripe of infinitesimal/greater quantum 3-D extension' is the physical thing; the 'notional line' which you 'reduce' it to in your mind/graph is merely the notional axis of 3-D extension along the length of the thing you painted etc. There is no 'dimensionless surfaces, points or lines', they are pure abstraction and philosophical notions of direction/extension of the real 3-D objects you are 'abstracting' from. Really, I have finished editing typos. I can't stay. So if I haven't got the essentials across to you I am sorry.

PS: arfa, I have read recent posts from you on 'time' and 'entanglement' in other threads. I am impressed. You have what it takes to break your 'mainstream conditioning' and think for yourself. It has been a pleasure to note it. It is with that increasingly justified respect I have for your original thought capacity in mind that I will respectfully leave you to ponder further (if you have a mind to) what I have pointed out objectively without fear or favor for your benefit as well as mine or the interests of science and humanity in general. Cheers!

 

[rpenner]

Trippy banned me for 14 days

What makes you think that Trippy's action was unknown to James R or the anyone in community of moderators when it was widely visible and ascribed to Trippy at the top of the "Ban List" page [link copied from page header] during the duration of your temporary absence? Are you really trying to correct a perceived failing of James R or are just grousing in a passive-aggressive manner?

I pointed out his transgressions

At best you could point out certain actions and argue that they were transgressions. Not everyone need agree. Likewise, not everyone need agree that your side-stepping approved channels to the moderators, super-moderators and admins was responsible and admirable behavior on your part.

as the "Philosophy of Maths' in the title implies

Philosophy is more than ignorant bloviation in favor of continued ignorance. Declaring the topic to be philosophy of math doesn't fairly allow you to ignore math arguments.

And speaking of infinities and the absurdities which start the maths of such, I draw your attention to this from rpenner to hansda:

Do you think, $$\frac{9}{10^\infty} = 0 $$ ?

It doesn't matter what I believe. It matters which mathematical axioms I choose to work with. Different axioms, different results. In this case $$\infty$$ as a quasi-number is only defined in one of the two systems I talk about and in that system $$T_{\infty+1} = \frac{1}{10} T_{\infty} = 0$$.

But neither of those two systems is necessary to everything in [post=3164396]Post #915[/post] being correct.

Now I ask you, James R, everyone, how can maths system 'choose' to start/base their following logics and treatments and results from such an obviously absurd 'axiom'?

Consider: Once an "$$\infty$$" has been invoked into an expression, how can anyone logically, physically (or even philosophically for that matter!) possibly pretend that they can "add 1" to such a thing?

Context restored.

As you see, it was hansda who began (without foundation) to claim to do arithmetic with the symbol $$\infty$$. As I have said many times, $$\infty$$ is not part of the natural numbers or real numbers and here I assume he is meaning to invoke the extended real numbers as treated in analysis which is the most common symbolic system that purports to allow some arithmetic with the symbol $$\infty$$. In such notation, $$\lim_{n\to\infty} x_n = \infty$$ just says there exists a function from the reals to the natural numbers, $$f(r)$$, such that for every real number $$r$$, for every natural number larger than $$f(r)$$, the real number $$x_n$$ is larger than $$r$$. Thus because $$\lim_{n\to\infty} ( x_n + 1 ) = \infty \quad \leftrightarrow \quad \lim_{n\to\infty} x_n = \infty$$ and $$\lim_{n\to\infty} 10^{x_n} = \infty \quad \leftrightarrow \quad \lim_{n\to\infty} x_n = \infty$$, a shorthand notation of $$\infty + 1 = \infty$$ and $$10^{\infty} = \infty$$ has rigorous foundation. Similarly, $$\frac{1}{\infty} = 0$$ but NOT $$\frac{1}{0} = \infty$$ or $$\infty - \infty = 0$$.

This is actually covered in the reference that hansda tried to invoke as an authority that supported his claims: See http://science.kennesaw.edu/~plaval/math4381/seq_limthm.pdf Theorem 315, included proof, and Remark 316 except for part 3(d) which does not follow from Theorem 315 and has counterexamples.

Nothing in [POST=3164396]post #915[/POST], however, requires working in a non-standard number system like hansda introduced or doing arithmetic with the symbol $$\infty$$.

 

[Undefined]

What makes you think that Trippy's action was unknown to James R or the anyone in community of moderators when it was widely visible and ascribed to Trippy at the top of the "Ban List" page [link copied from page header] during the duration of your temporary absence? Are you really trying to correct a perceived failing of James R or are just grousing in a passive-aggressive manner?
At best you could point out certain actions and argue that they were transgressions. Not everyone need agree. Likewise, not everyone need agree that your side-stepping approved channels to the moderators, super-moderators and admins was responsible and admirable behavior on your part.

A fine example of your own failures as 'mod' is still on the net. So your irrelevant opinions are neither here nor there. Especially if you haven't a clue what went down in these latest incidents (except by Trippy's 'version' of the events and your own assumptions and beliefs in Trippy's 'infallibity'?).

And I was prevented from PMing the latest incident facts to James because Trippy banned me. How can one approach the Admin etc whilst I was banned? Get real and stop with the spurious red herrings.


And once I PM James he can speak for himself; he doesn't need you to speak for him or create another 'spurious version' of the actual incidents involved.

Philosophy is more than ignorant bloviation in favor of continued ignorance. Declaring the topic to be philosophy of math doesn't fairly allow you to ignore math arguments.

Context restored.

As you see, it was hansda who began (without foundation) to claim to do arithmetic with the symbol $$\infty$$. As I have said many times, $$\infty$$ is not part of the natural numbers or real numbers and here I assume he is meaning to invoke the extended real numbers as treated in analysis which is the most common symbolic system that purports to allow some arithmetic with the symbol $$\infty$$. In such notation, $$\lim_{n\to\infty} x_n = \infty$$ just says there exists a function from the reals to the natural numbers, $$f(r)$$, such that for every real number $$r$$, for every natural number larger than $$f(r)$$, the real number $$x_n$$ is larger than $$r$$. Thus because $$\lim_{n\to\infty} ( x_n + 1 ) = \infty \quad \leftrightarrow \quad \lim_{n\to\infty} x_n = \infty$$ and $$\lim_{n\to\infty} 10^{x_n} = \infty \quad \leftrightarrow \quad \lim_{n\to\infty} x_n = \infty$$, a shorthand notation of $$\infty + 1 = \infty$$ and $$10^{\infty} = \infty$$ has rigorous foundation. Similarly, $$\frac{1}{\infty} = 0$$ but NOT $$\frac{1}{0} = \infty$$ or $$\infty - \infty = 0$$.

This is actually covered in the reference that hansda tried to invoke as an authority that supported his claims: See http://science.kennesaw.edu/~plaval/math4381/seq_limthm.pdf Theorem 315, included proof, and Remark 316 except for part 3(d) which does not follow from Theorem 315 and has counterexamples.

Nothing in [POST=3164396]post #915[/POST], however, requires working in a non-standard number system like hansda introduced or doing arithmetic with the symbol $$\infty$$.

You would know all about 'bloviating' and 'blogging' (and the unsavory rest) from your own performance/history at that unfortunate site. So don't go there, hey?

As for the context of the $$\infty + 1$$ expression, it's part and parcel of the Hilbert's Hotel examples/exercises regarding the treatments/assumptions. That was the context, any reference made via quote earlier was merely the device to bring up that particular expression for further consideration.

So, rpenner, without any more bloviation from you, do you or do you not subscribe to the Hilbert's Hotel arguments etc regarding infinities and additional guests coming in to any already infinitely full hotel?

If you do, then the context and the observations are correct, and I have identified where the fault lay with the 'maths' in that case. If you do not, then you either agree with me where the axioms/treatments/arguments are non-sequiturs as explained, or you can identify where you think the non-sequiturs are in Hilbert Hotel and other such like absurdly philosophical manipulations while calling them 'rigorous maths'.

Back tomorrow to see.

 

[arfa brane]

There is no 'dimensionless surfaces, points or lines', they are pure abstraction and philosophical notions of direction/extension of the real 3-D objects you are 'abstracting' from.

This, according to you, is some kind of problem.

Humans have been abstracting for a long time; abstraction and imagination by humans is why civilisation developed--humans began imagining the future beyond the next few days, or the next hunt. Abstraction isn't a problem; numbers are not a problem, nor is drawing lines which are not one-dimensional on a surface and imagining the lines are one-dimensional and so can intersect at a zero-dimensional point.

This is despite the fact that you can't draw something with zero dimensions and that two lines drawn on the ground will intersect (if they do) at a region which has more than zero dimensions. It's very simple: just imagine that the lines are one dimensional.
As I state, humans can imagine and abstract. That we can count camels is a testament to this since the numbers used during counting don't exist physically whereas the camels do, although these days a mathematician might insist the camels are rational rather than whole. This "counting camels" is a philosopical abstraction an "overlay" as you put it, based on axioms. Without the axioms, no camels get counted.

Now you are arguing that abstraction is problematic and leads to "absurdities", but you haven't shown anything, all you've done is insist there's some kind of crisis. This to me implies that humans must have based the growth of civilisation on a mistaken assumption--and yet here we are with civilisation intact for the most part. So what, then, is wrong with the axioms we continue to use, such as: there is no largest natural number, and the smallest is 1?

I can't see a problem here.

As for the context of the $$\infty + 1$$ expression, it's part and parcel of the Hilbert's Hotel examples/exercises regarding the treatments/assumptions. That was the context, any reference made via quote earlier was merely the device to bring up that particular expression for further consideration.

Can you show me an analysis of Hilbert's problem that uses $$\infty + 1$$? Or that treats $$\infty$$ as a number with a value?
Certainly the concept of an infinite "number" of rooms and guests is used to analyse Hilbert's Hotel, but I can't recall ever seeing $$\infty +1$$ used. The problem is really about finding what's known as a bijection between rooms and guests, as infinite sets.

 

[rpenner]

As for the context of the $$\infty + 1$$ expression, it's part and parcel of the Hilbert's Hotel examples/exercises regarding the treatments/assumptions. That was the context, any reference made via quote earlier was merely the device to bring up that particular expression for further consideration.

Can you show me an analysis of Hilbert's problem that uses $$\infty + 1$$? Or that treats $$\infty$$ as a number with a value?
Certainly the concept of an infinite "number" of rooms and guests is used to analyse Hilbert's Hotel, but I can't recall ever seeing $$\infty +1$$ used. The problem is really about finding what's known as a bijection between rooms and guests, as infinite sets.

Indeed, the only property of "the infinite" used in the Hilbert Hotel metaphor is the unending progression of larger natural numbers and from that it is easy to prove that there can be a bijection between all the natural numbers and a subset of natural numbers.

http://us.metamath.org/mpegif/unbnn2.html
http://us.metamath.org/mpegif/unben.html
http://us.metamath.org/mpegif/prminf.html
"If A is a subset of the set of all natural numbers and for every natural number x there exists at least one member of A which is greater than or equal to x then it follows that there is a bijection between the natural numbers and the subset A. Thus it follows that there is, for example, a bijection between natural numbers and primes."

But just because we establish that we are doing arithmetical mappings of arbitrary natural numbers does not establish that we are doing arithmetic on anything other than natural numbers. $$\infty$$ is not a natural number so your contention that $$\infty + 1 = \infty$$ is part of the Hilbert Hotel discussion appears to be without basis. The basis for $$\infty + 1 = \infty$$ (in the context of unending sequences that tend to infinity) was a part of [post=3171869]post #1093[/post] (expanding on a link to hansda's reference originally in [post=3170728]post #1081[/post]) that you quoted but did not respond to). Similar statements exist in the domain arithmetic of the cardinal numbers, but don't use the symbol $$\infty$$ because of Cantor's paradise of an infinite number of distinct infinities. Likewise, $$\infty + 1 \gt \infty$$ is analogous to statements made in the arithmetic of the hyperreals, surreals and ordinal numbers, which also don't use the symbol $$\infty$$. Different ideas, different maths as I have been explaining to hansda for some time now.

But if you aren't concrete about your ideas then you can't explore their ramifications.

// Added in edit:

Perhaps the idea that the Hilbert Hotel uses the concept of $$\infty + 1 = \infty$$ comes from the fact that discussion usually begins with a full hotel and the need is made to accommodate a new guest. But that is demonstrating from the unending nature of the set of natural numbers that it follows that a bijection exists between the set of natural numbers and the union of natural numbers and a singleton consisting of a new element -- but that's just the definition of the arithmetic of infinite cardinals. Specifically $$\textrm{card}\left( \left( \left{0\right} \times \mathbb{N} \right) \cup \left( \left{1\right} \times \left{0\right} \right) \right) = \textrm{card}\left( \mathbb{N} \right) \quad \Rightarrow \quad \omega +_{\tiny \textrm{card}} 1 = \omega$$. Where cardinal arithmatic is based on the equivalence relationship between sets that can be paired off by bijections. It's not about "infinity" per se, but about any particular cardinality of an infinite set. So the symbol $$\infty$$ isn't used.

To the extent Undefined argues that the Hilbert Hotel uses the concept of $$\infty + 1 = \infty$$, he gets it backwards. If anything the Hilbert Hotel illustrates the difference between cardinal arithmetic of the finite and cardinal arithmetic of the infinite by working with explicit bijections between the natural numbers (room numbers) and the disjoint union of sets of cardinality less than or equal to the cardinality of the natural numbers (guests) but it only uses the fact that the natural numbers are unending.

do you or do you not subscribe to the Hilbert's Hotel arguments etc regarding infinities and additional guests coming in to any already infinitely full hotel?

To what specific arguments do you refer? You aren't referring to an actual hotel, but instead a metaphor made for the purpose of introducing the concept of bijections between the natural numbers and subsets of the natural numbers. Thus, as I am educated in set theory, I need never refer to any arguments made about the fictional hotel but immediately make the cognitive leap to the actual point about the set of all natural numbers, without getting bogged down in the mental baggage of actual hotel experience.

If you have let yourself get distracted by some trivia of the metaphor that is not fundamental to the actual mathematics, then that would be a learning disability on your part not an objectionable argument in the popular literature.

If you do,

I have no problem with the metaphor of a hotel with numbered rooms and guests moving between them to illustrate a one-to-one mapping. But I don't reason with the metaphor but with the underlying concept.

then the context and the observations are correct, and

That doesn't follow, since the $$\infty + 1 = \infty$$ that you complain in this post is a result from analysis and sets of the reals, not natural numbers, that are unbounded above.

I have identified where the fault lay with the 'maths' in that case.

Specific citation required to support your claims.

 

[Undefined]

This, according to you, is some kind of problem.

Humans have been abstracting for a long time; abstraction and imagination by humans is why civilisation developed--humans began imagining the future beyond the next few days, or the next hunt. Abstraction isn't a problem; numbers are not a problem, nor is drawing lines which are not one-dimensional on a surface and imagining the lines are one-dimensional and so can intersect at a zero-dimensional point.

This is despite the fact that you can't draw something with zero dimensions and that two lines drawn on the ground will intersect (if they do) at a region which has more than zero dimensions. It's very simple: just imagine that the lines are one dimensional.
As I state, humans can imagine and abstract. That we can count camels is a testament to this since the numbers used during counting don't exist physically whereas the camels do, although these days a mathematician might insist the camels are rational rather than whole. This "counting camels" is a philosopical abstraction an "overlay" as you put it, based on axioms. Without the axioms, no camels get counted.

Now you are arguing that abstraction is problematic and leads to "absurdities", but you haven't shown anything, all you've done is insist there's some kind of crisis. This to me implies that humans must have based the growth of civilisation on a mistaken assumption--and yet here we are with civilisation intact for the most part. So what, then, is wrong with the axioms we continue to use, such as: there is no largest natural number, and the smallest is 1?

I can't see a problem here.Can you show me an analysis of Hilbert's problem that uses $$\infty + 1$$? Or that treats $$\infty$$ as a number with a value?
Certainly the concept of an infinite "number" of rooms and guests is used to analyse Hilbert's Hotel, but I can't recall ever seeing $$\infty +1$$ used. The problem is really about finding what's known as a bijection between rooms and guests, as infinite sets.

Hi again, arfa. Sorry, I am still rushed, so as briefly as I can...
Are you claiming that "camels" don't 'exist' until you have a 'number axiom' to 'count' them? Relax, I know that's not what you claim...it's just the effective implication of your counterargument when it's boiled down to the essentials. Abstraction and imagination are philosophical overlays. I'm glad we agree on that at least. Now consider: Camels exist in reality (as do all sorts of 'entities' which we may label as 'units' of some kind which either have objective existence or are related to our 'pattern recognition/disrmination' facility of our brain-mind). Numbers do not exist in reality. Hence we are making philosophical overlays when we construct a 'number system' abstraction.

The question of what is maths and what is the philosophy behind the maths is what this discussion is about. I do not argue against the 'usefulness' of imagination or abstractions or philosophy concepts/overlays. But I am pointing out where the current maths and its axioms are founded on things which are NOT mathematical entities or concepts to begin with. Hence the potential for continued 'undetermined' and 'undefined' and 'invalid expressions' etc etc from the current maths as founded on he philosophical overlays/notions I pointed out before.

The aim of my axiomatic re-examination and re-starting is to produce a maths which does NOT output such non-sequiturs and undetermineds etc etc that the currently formulated axioms, if followed rigorously, will end up in all too often. Now you can see that is NOT what complete and consistent and rigorous system of logics should be 'outputting' if the axioms WERE adequate to the task in the first place? One obvious example of the problem is the 'like/like' construction from the current axioms/treatments which FAIL when the expression '0/0' arises. Please read my post above to James, where I point out that if the '0' is a number then the '0/0' expression should be as valid as the '9/9' expression. See the current inadequacies of the current maths/axioms as being practiced from the original axioms that are NOT eventually capable of making sense of that '0/0' (and other sorts of equally axiomatically-incapable examples) expressions?

We use the "1" and the "0" but do not currently realize that "1" and "0" actually logically, philosophically AND physically ALSO REPRESENT 'superposition/transition' indicator/symbol in certain/many contexts when one looks deep down beneath all the 'wall of maths and text' in which they are manipulated/involved. My improved axioms would recognize this from the get-go, and all these and other undetermined/problematic treatments and outcomes would NOT arise, but be capable of sensibly and consistently being handled without ad hoc 'fixes' and 'rechoosing of expedient axioms as you go along' as happens at present.


Re Hilbert Hotel aspect, [/b]please note that I restricted my argument advisedly to the part of the expression that says "$$\infty + 1$$" ONLY.[/b]

It is that 'adding to infinity' concept that is the problem; not whether or not infinity is a number or not. The right side of the 'overall equation' in the original quote (ie, the "= "$$\infty$$ part) is neither here nor there, as the left side (ie, ""$$\infty + 1$$") is a non-sequitur in any sense, even in philosophy, so anything that follows or depends on that is doomed to nonsense, and doesn't even get a look in (hence one may ignore the "=$$\infty$$" (and the overall equation altogether), as merely a philosophical abstraction having no real meaning in any sense since the left side is just as meaningless/invalid to begin with.

See? The purely philosophical notion that one can invoke an infinity and THEN 'add' to it is pure absurdity. I trust that point has got across now? That sort of nonsense is invoked in Hilbert Hotel treatments of 'infinities' and 'additions' to same. Therein, any ARBITRARY differentiations of the 'room numbers' into 'odd and even' is a trivial philosophical exercise because taken together the original 'infinity of rooms' will ALREADY BE COMPLETE as stated (unless NOW you and others cant to claim that it is a "process". No? Good.).

So any of that sort of 'mapping' and 'bijection' etc manipulations are purely philosophical distinctions/exercises, using equally philosophical (remember the 'camels' are real; while the 'numbers' and LABELS are philosophical, as above explained?).

The whole mess is predicated from philosophical 'points' and 'infinities being arbitrarily invoked and dissected equally philosophically when it is looked at closely. Hence the ned to re-jig the axiomatic foundations to avoid from the get-go all the PHILOSOPHY INSIDIOUSLY infesting the supposedly rigorous maths system all the way through to 'undefined' and 'philosophical exercise' outputs and fixes by which the current axioms/practice have doomed maths (and by association) physics to INCOMPLETION because it is PHLOSOPHY at ROOT, not rigorously REAL mathematics/physics as it has evolved to date.

Mate, the rest I have to save for my complete and consistent maths-physics ToE publication. I can say no more than I have pointed out here and elsewhere already, because plagiarism/gazzumping is a real risk at this stage. I will leave you to work the rest out for yourself, and if you succeed in 'getting it' before you read my book, then more power to your original thinking, arfa! Thanks for the polite/interesting conversations. Cheers and bye for now (I will briefly respond to rpenner before logging out).

 

[Undefined]

Indeed, the only property of "the infinite" used in the Hilbert Hotel metaphor is the unending progression of larger natural numbers and from that it is easy to prove that there can be a bijection between all the natural numbers and a subset of natural numbers.

http://us.metamath.org/mpegif/unbnn2.html
http://us.metamath.org/mpegif/unben.html
http://us.metamath.org/mpegif/prminf.html
"If A is a subset of the set of all natural numbers and for every natural number x there exists at least one member of A which is greater than or equal to x then it follows that there is a bijection between the natural numbers and the subset A. Thus it follows that there is, for example, a bijection between natural numbers and primes."

But just because we establish that we are doing arithmetical mappings of arbitrary natural numbers does not establish that we are doing arithmetic on anything other than natural numbers. $$\infty$$ is not a natural number so your contention that $$\infty + 1 = \infty$$ is part of the Hilbert Hotel discussion appears to be without basis. The basis for $$\infty + 1 = \infty$$ (in the context of unending sequences that tend to infinity) was a part of [post=3171869]post #1093[/post] (expanding on a link to hansda's reference originally in [post=3170728]post #1081[/post]) that you quoted but did not respond to). Similar statements exist in the domain arithmetic of the cardinal numbers, but don't use the symbol $$\infty$$ because of Cantor's paradise of an infinite number of distinct infinities. Likewise, $$\infty + 1 \gt \infty$$ is analogous to statements made in the arithmetic of the hyperreals, surreals and ordinal numbers, which also don't use the symbol $$\infty$$. Different ideas, different maths as I have been explaining to hansda for some time now.

But if you aren't concrete about your ideas then you can't explore their ramifications.

// Added in edit:

Perhaps the idea that the Hilbert Hotel uses the concept of $$\infty + 1 = \infty$$ comes from the fact that discussion usually begins with a full hotel and the need is made to accommodate a new guest. But that is demonstrating from the unending nature of the set of natural numbers that it follows that a bijection exists between the set of natural numbers and the union of natural numbers and a singleton consisting of a new element -- but that's just the definition of the arithmetic of infinite cardinals. Specifically $$\textrm{card}\left( \left( \left{0\right} \times \mathbb{N} \right) \cup \left( \left{1\right} \times \left{0\right} \right) \right) = \textrm{card}\left( \mathbb{N} \right) \quad \Rightarrow \quad \omega +_{\tiny \textrm{card}} 1 = \omega$$. Where cardinal arithmatic is based on the equivalence relationship between sets that can be paired off by bijections. It's not about "infinity" per se, but about any particular cardinality of an infinite set. So the symbol $$\infty$$ isn't used.

To the extent Undefined argues that the Hilbert Hotel uses the concept of $$\infty + 1 = \infty$$, he gets it backwards. If anything the Hilbert Hotel illustrates the difference between cardinal arithmetic of the finite and cardinal arithmetic of the infinite by working with explicit bijections between the natural numbers (room numbers) and the disjoint union of sets of cardinality less than or equal to the cardinality of the natural numbers (guests) but it only uses the fact that the natural numbers are unending.

To what specific arguments do you refer? You aren't referring to an actual hotel, but instead a metaphor made for the purpose of introducing the concept of bijections between the natural numbers and subsets of the natural numbers. Thus, as I am educated in set theory, I need never refer to any arguments made about the fictional hotel but immediately make the cognitive leap to the actual point about the set of all natural numbers, without getting bogged down in the mental baggage of actual hotel experience.

If you have let yourself get distracted by some trivia of the metaphor that is not fundamental to the actual mathematics, then that would be a learning disability on your part not an objectionable argument in the popular literature.

I have no problem with the metaphor of a hotel with numbered rooms and guests moving between them to illustrate a one-to-one mapping. But I don't reason with the metaphor but with the underlying concept.
That doesn't follow, since the $$\infty + 1 = \infty$$ that you complain in this post is a result from analysis and sets of the reals, not natural numbers, that are unbounded above.
Specific citation required to support your claims.

All well and good regurgitation from the current maths assumptions/treatments/practices textbook. Meanwhile you miss the point made.

If you had properly read and understood my earlier post to James (and now arfa), you would have noted that in my argument/example I used ONLY the "$$\infty + 1$$" part of the expression "$$\infty + 1 = \infty$$". I did that ADVISEDLY because that alone demonstrated the absurdity and the philosophical (and in no way mathematical) nature of that starting notion that one can "add to infinity" and pretend to proceed to any sort of 'rigorous' mathematical outcomes/treatments therefrom.

See? All your bloviation and blogging from current maths understandings/practices are made MOOT and mere absurd sophistry by that point, and so can be ignored entirely. The fact that the Hilbert Hotel exercises indulge in ARBITRARY differentiation of LABELS (room numbers), and then proceed to map and biject etc between these arbitrary distinctions is neither here nor there, and merely NUMEROLOGY and/or PHILOSOPHICAL MATH-TURBATION when closely examined.

Can you not see that the invocation of any infinity (infinity of ROOMS already there in Hilbert Hotel, irrespective of any LABELING/NUMBERING with philosophical/abstract number construct overlays) immediately and sensibly and logically (and also philosophically even) BOUNDS the initial expression of that infinity of rooms in EVERY SENSE, and PRECLUDES EFFECTIVELY any further 'adding' to it or to arbitray dissection which IMMEDIATELY CHANGES the original infinity such that you can no longer sensibly USE the originally invoked ROOMS INFINITY, because you have ARBITRARILY invoked other TOTALLY DIFFERENT infinities when attaching/assuming ROOM NUMBERS/LABELS which can then be trivially dissectied into odd and even to allow your further numerological/math-turbational exercises which essentially bil down to PHISOLOSOPHICAL exercises having nothing to contribute/offer rigorous mathematics/axioms AT ALL.

They are 'ad hoc' numerological/philosophical/abstract 'overlays' which confuse rather than enlighten....all because you started out from the philosophical axioms of 'points' etc and end up inevitably math-turbating like that while pretending it is 'mathematics'. It can NOT be mathematics unless the absurd philosophy 'axioms' are ROOTED OUT from the start. Only then will all these nonsensical absurdities and undefineds and undertermineds and all the rest I pointed out to James/arfa above can be AVOIDED and handled properly with the proper REAL axioms. Improve the axioms and you improve the chances for real complete rigorous mathematics/physics. That is where I am coming from. So please no more kneejerking from INCOMPLETE maths-as-is textbooks/understandings/practices, and just follow the NEW INFORMATION/PERSPECTIVE which indicates a review of the axioms/maths-as-is is NECESSARY for the ever-questioning scientific method to arrive at the completion of both the maths and the physics models/ToEs.

As I mentioned to arfa, I can't say any more on these matters at this stage, for the reasons stated to him. So goodbye for now, rpenner; and I wish you good luck and good thinking in future. No hard feelings here, irrespective. Life is too short, hey? Cheers, robert, everyone! Please forgive the typos, I haven't time to go back and deal with them. Enjoy your reality, wherever you are in life's trajectory.

 

[arfa brane]

But I am pointing out where the current maths and its axioms are founded on things which are NOT mathematical entities or concepts to begin with.

I haven't seen much of this "pointing out" though.

You may have mentioned that the concept of a point with no dimensions is problematic. I can't personally see what the problem is: although a zero-dimensional object is unphysical, you can still label it.
So mathematically, points are just like camels which have had all their physical dimensions "suppressed" abstractly (since suppressing dimensions physically is hard to do, and entirely unnecessary).

And you seem to have issues with zero itself and infinity. Certainly zero is a number which is different from other nonzero numbers, and infinity isn't really a number.
Do you think nobody else has noticed that these particular mathematical entities have to be treated specially? You say this:

So any of that sort of 'mapping' and 'bijection' etc manipulations are purely philosophical distinctions/exercises

. Are you claiming that set theory needs to be "re-jigged"?
If you are, you will definitely need something more rigorous than "purely philosophical distinctions". Mathematics is a philosophy, so it looks like you're saying "purely mathematical distinctions". If you don't mean that, then you should be more precise about what you think the problems are, that is, show how assuming there are infinite sets leads to absurdities, how assuming lines are one-dimensional means there is a problem when they intersect, and so on.

As far as mathematics being "doomed to incompletion", this doesn't really make sense. Mathematics describes axiomatically complete (though unphysical) systems, but mathematics is an incomplete philosophy; if it was complete there would be nothing left to discover about it, which clearly does not follow for the current state of affairs.

 

[Undefined]

I haven't seen much of this "pointing out" though.

You may have mentioned that the concept of a point with no dimensions is problematic. I can't personally see what the problem is: although a zero-dimensional object is unphysical, you can still label it.
So mathematically, points are just like camels which have had all their physical dimensions "suppressed" abstractly (since suppressing dimensions physically is hard to do, and entirely unnecessary).

And you seem to have issues with zero itself and infinity. Certainly zero is a number which is different from other nonzero numbers, and infinity isn't really a number.
Do you think nobody else has noticed that these particular mathematical entities have to be treated specially? You say this: . Are you claiming that set theory needs to be "re-jigged"?
If you are, you will definitely need something more rigorous than "purely philosophical distinctions". Mathematics is a philosophy, so it looks like you're saying "purely mathematical distinctions". If you don't mean that, then you should be more precise about what you think the problems are, that is, show how assuming there are infinite sets leads to absurdities, how assuming lines are one-dimensional means there is a problem when they intersect, and so on.

As far as mathematics being "doomed to incompletion", this doesn't really make sense. Mathematics describes axiomatically complete (though unphysical) systems, but mathematics is an incomplete philosophy; if it was complete there would be nothing left to discover about it, which clearly does not follow for the current state of affairs.

Hi arfa. My guest has been delayed a few minutes, so I came in to edit typos and saw your reply.

Mate, it's no good trying to get me to explain more than I have already done here and elsewhere. Like I said, at this stage towards publication, the risk of plagiarism/gazzumping is all too real. There is much more depth and complete explanations in my maths-physics ToE to be published. Until then I have no more to add. I hope you understand that I am not evading or disrespecting you or anyone? Thanks.

But I can briefly address your above items and then leave it at that:

Hehehe. Just try placing/locating 'infinite camels' on a 'line', and you'll see that 'infinite points' and 'infinite camels' are altogether different things. Camels real; points not real. The rest of the problems I have explained already, especially as it relates to "0" and "point" and INFINITESIMALS of effectiveness which any system/process/construct must include as an inherent part of its 'logics/construction/processes' if it is not to lead to abstract nonsense (like your above abstract notion/attempt to place/locate an infinity of camels on a line as you would an infinity of points? See it now? ).

If zero is a number (like "1" ) representing a superposition state of transition, then there is no worries, and the axioms would be able to handle it like any other 'like/like' expression. That is the point of re-jigging the axioms, so that it all comes out IN CONTEXT and does not confuse the issue like the abstractions/nonsense undefineds and etc do, which prevent completion of both the physics and the maths.

Also please try not to forget that point I made about infinitesimal effectiveness step which makes the transition/superposition state between two different state (including different 'infinities' etc). The whole thrust is to make the axioms more real from the get-go such that there won't be all these (at root) numerology/philosophy/abstraction based dead-ends which make the maths doomed to incompleteness (because it ends up as PHILOSOPHY not maths, see? That is why it is currently doomed to incompleteness, because philosophy and 'un-reality' fantasy worlds we play in but do not have any connection with reality. Sure, we can go on forever with the fantasy worlds constructs, but it doesn't complete the reality maths/physics construct our ToE must address).

Unlike you and others, I am no defeatist when it comes to humanity's capability to complete the maths/physics theory consistent and real to avoid all the false starts/ends which are mired in insidious philosophy/numerology/abstract concepts which have no place in the FINAL COMPLETE maths/physics explanation/constructs/models. Just because you and others are reticent to tackle it as I have, it does not mean that such completion is impossible. I am busy proving it is not impossible, by letting the complete and consistent maths/physics ToE speak for itself when published. I have let too many 'cats out of the bag' as it is. But until you see the whole thing when published, it may seem impossible to you. But don't despair, humanity is not all in thrall to 'un-real' thinking to the exclusion of the reality around us which informs our thinking if we let it, without overlaying too many abstraction and philosophical misdirections.

If you really want to get a better idea of all the subtleties involved, you could do worse than going back and finding the relevant posts of mine (and others too!) and studying them again with an original bent of mind. Study them as you have done your current 'texts', for in these new discussions BEHIND the maths-as-is, there are things which your current 'incomplete maths' texts/practices cannot ever encompass unless the axioms are changed to reflect the reality. Leave the philosophy and the fantasy effectively trying to equate 'infinite points' with 'infinite camels' whenever you try to 'find/place' such on an abstract line from "0" to "1" construct/process (for, as MD might say: "That way lay madness!"). Gotta go. Cheers.