# Are Infinitesimals Really Numbers?

Discussion in 'Physics & Math' started by TruthSeeker, Mar 29, 2017.

1. ### TruthSeekerFancy Virtual Reality MonkeyValued Senior Member

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We know that infinity is not a number and it cannot be constructed. We also know that infinitesimals cannot be constructed and successor relations fall apart because there are always more infinitesimals in between two infinitesimals. An infinitesimal cannot be defined as a completed infinite totality, it cannot be encapsulated. If this is the case, how can we define infinitesimals as numbers? Wouldn't they just be a form of infinity?

3. ### Dr_ToadIt's green!Valued Senior Member

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Why do you want to complicate something so simple? Did you not take any calculus class, or did you die off in geometry?

5. ### TruthSeekerFancy Virtual Reality MonkeyValued Senior Member

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Have you ever done any real math? (haha)
Of course I have done calculus, but you can't construct infinitesimals with calculus, the best we have been able to do, as far as I am aware of, is with cauchy sequences and dedekind cuts. Do you even know what I am talking about?

7. ### Dr_ToadIt's green!Valued Senior Member

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Yes, but what the hell are you asking? An infinitesimal isn't constructed. It's an artifact (or tool) of the math you use...

8. ### TruthSeekerFancy Virtual Reality MonkeyValued Senior Member

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I guess you haven't studied intuitionistic logic or intuitionistic math... The basic idea is that if you don't have a formula to define a number you can't prove that it exists. There is no formula that can define an infinitesimal number, therefore, we cannot prove that infinitesimals exist.

https://plato.stanford.edu/entries/intuitionism/

9. ### Dr_ToadIt's green!Valued Senior Member

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OK. You win, enjoy your arrogance.

Until someone like rpenner can correct my my misunderstanding. Or yours.

Edit: As I said, what are you asking for in the OP beyond the awesomeness and depth of your thought in the statement of it?

10. ### The GodValued Senior Member

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would it help.

The article is quite an eye opener on infinitesimals..
https://en.wikipedia.org/wiki/Infinitesimal

11. ### TruthSeekerFancy Virtual Reality MonkeyValued Senior Member

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Yeah, that suggests I was correct in my thinking. I guess I'm just used with people referring to infinitesimals as though they are numbers.

12. ### originHeading towards oblivionValued Senior Member

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Who thinks of infinitesimals (or infinity) as numbers?

13. ### arfa branecall me arfValued Senior Member

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Infinitesimals are a set of numbers smaller than any other number. Their inverse is the set of numbers which are finite, but larger than any other number.

14. ### TruthSeekerFancy Virtual Reality MonkeyValued Senior Member

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But infinitesimals are, by definition, uncountably infinite. They are not countable. That's the whole point of the real numbers forming a continuum, it has a cardinality larger than the set of any countable number or sequence of numbers. Infinity and infinitesimals are both undefined quantities. They can't be counted, therefore, they are technically not "numbers", they are concepts that refer to uncountable quantities. Infinitesimal means precisely that they are too small to ever be counted.

15. ### TruthSeekerFancy Virtual Reality MonkeyValued Senior Member

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People who don't study math, I guess.

16. ### originHeading towards oblivionValued Senior Member

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Yes, I was wrong. Arfa was correct. Infinitesimals approach infinity they are finite.

17. ### TruthSeekerFancy Virtual Reality MonkeyValued Senior Member

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If they are finite, then there must be a formula that enables their construction. Is there such a formula? What is it?

18. ### QuarkHeadRemedial Math StudentValued Senior Member

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Are they? In fact, what does this mean? Does it mean that between any 2 arbitrarily "close" ordinals there are uncountably many so-called "infinitesimals"? Then why can they not just be Real numbers? Both Cauchy and Dedekind proved they are.in fact.

No. Countable infinity is a very well-defined property of, say, the Natural numbers.
I cannot parse this - a set is said to be uncountable iff it is "too large" be put into a one-to-one correspondence with a subset of the Natural numbers. If the cardinality of a set is "infinitesimally small", how can it's cardinality be uncountably "large"?

19. ### arfa branecall me arfValued Senior Member

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Something that might help (if you can read a bit of math): http://quantum-gravitation.de/media/f483977d7e58f426ffff824fffffffef.pdf

The author states: "In addition to the usual undefined binary predicate ∈ of set theory we adjoin a new undefined unary predicate standard.
The axioms of IST are the usual axioms of ZFC plus three others, which we will state below.
All theorems of conventional mathematics remain valid.
No change in terminology is required. What is new in internal set theory is only an addition, not a change. We choose to call certain sets standard (and we recall that in ZFC
every mathematical object-a real number, a function, etc.-is a set), but the theorems of conventional mathematics apply to all sets, nonstandard as well as standard"

That is, they just take R, the real numbers, and add some extra (axiomatic) definitions such that a new relation is defined between pairs. One view of this relation is where the author states: "Two real numbers x and y are called infinitely close, denoted by $x \simeq y$ in case x - y is infinitesimal."

hansda likes this.
20. ### The GodValued Senior Member

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More or less you can take this as an 'adjective' in general. But in rigorous maths in calculus an integration of infinite infinitesimals may lead to a finite number.

21. ### hansdaValued Senior Member

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If you follow the wiki article on numbers https://en.wikipedia.org/wiki/Number ; infinity and infinitesimals are included there.

22. ### arfa branecall me arfValued Senior Member

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We're supposed to think of infinitesimals and their inverses as numbers that can never be written down, because they're too small (requiring, say, a decimal representation that would have a nearly infinite number of zeros after the decimal point), or too large (requiring a decimal representation that would take "forever" to write down); there aren't enough sheets of paper now, or in any future, that will accommodate this.

Albeit, a somewhat prosaic explanation.

23. ### someguy1Registered Senior Member

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The standard real numbers do not contain any infinitesimals.

There are some nonstandard models of the real numbers that do contain infinitesimals, but they have problems. They are not complete. They are full of holes. They contain Cauchy sequences that do not converge.

If a number field contains even a single infinitesimal, it's not Archimedean. And any non-Archimedean number field must necessarily be incomplete

For that reason, the hyperreals (and also the surreals) are not a satisfactory model of the continuum. If we know anything about the continuum, it's that it has no holes. The hyperreals superficially satisfy some people's intuitions of how the real numbers must be; but in fact the hyperreals are much more problematic than people realize.

Then someone mentioned the intuitionist line. As the hyperreals have the reals PLUS a lot of infinitesimals, the intuitionist line contains the reals MINUS all the noncomputable numbers.

In mathematical intuitionism or constructivism, a real number exists when its decimal expression can be produced by an algorithm. For example $\pi$ is computable as witnessed by the many algorithms you can find like Leibniz's series.

Since there are only countably many algorithms, the constructive real line has a lot of holes in it. The intermediate value theorem is false. The graph of a continuous function can pass right through the $x$-axis without intersecting it. What kind of continuum is that??

So here's this interesting thing:

* The intuitionist line is incomplete because it's too restrictive. Not enough points.

* The hyperreal line is incomplete because it's too permissive. It's got all the reals plus a cloud of infinitesimals around each real. It's got too many points.

* The standard reals are complete because they aren't too restrictive, and they aren't too permissive. They standard reals are "just right," like Goldilocks.

Last edited: Apr 6, 2017