Let's think about the first sentence. Assume it's axiomatically impossible for a program to have run forever in a universe that hasn't existed forever. Where is the importance of any metaphysical distinction, and between what?
The difference is between what mathematical objects one can claim exist (in some way) and what one cannot. If we limit ourselves to only those mathematical objects that we can construct, then this limits the kind of theorems we can prove and the kind of logical relationships we can use. Computation has rules that determine possibility, but it is still fundamentally constructive.
Yeah. I did a course recently on computational logic, we were told that a program can considered as some "chunk" of code, with a specific pre- and post-condition, something that can be formally written for a program P, like: {a}P{b}. The {a} and {b} are things that have to be true before and after P runs (and halts) on some computer. In the case of P running forever, the post-condition {b} must not ever be satisfied whatever it is. Does that allow some metaphysical context or what, since {b} can be anything? Suppose {b} includes that "P has run forever" be true, then P must halt to satisfy this post-condition (and whatever else is in {b}), if P halts it won't run forever though.
In the context of the computer world, you can write what is called an infinite loop. It merely means that the program will run until one of the following occurs: A human terminates the program, the computer is turned off, or the computer stops running due to some hardware failure.
"If you use division algorithmically, it's obvious that dividing 1 by 3" only one problem, algorithms result in only a finite collection of states no matter how long you run the algorithm.
which brings us to the justification that iterative successor operations evolve into an infinite collection. that is essentially what the axiom of infinity claims and contradicts any recursive logic we know.
have you considered running your algorithm according to dichotomy timing. this is a so called zeno machine.
It is not that I need to do more reading. It is that you need better understanding of your sources. I count the words "obvious" or "obviously" only appearing 23 times in Kunen's 1980 Set Theory and never in connection with axioms. On page xi, the first page of the Introduction, Kunen writes: Set theory is the foundation of mathematics. All mathematical concepts are defined in terms of the primitive notions of set and membership. In axiomatic set theory we formulate a few simple axioms about these primitive notions in and attempt to capture the basic “obviously true” set-theoretic principles. From such axioms, all known mathematics may be derived. However, there are some questions which the axioms fail to settle, and that failure is the subject of this book. Thus it is the principles which are “obviously true” (these are scare quotes), while the axioms are not-necessarily-obvious, but simple formulations which capture those principles. Why the scare quotes? Because what is obviously true depends on your preconceptions, not reality or some other source of universal truth. Chapter I, §3 explains that what principles are “obviously true” depends on your preconceptions and introduces the Finitists who do not accept infinite sets. Chapter I, §7, Theorem 7.4 is \( \vdash \neg \exists z \; \forall x \; \left( x \; \textrm{is an ordinal} \; \to x \in z \right) .\)Thus there cannot be a set of ALL the ordinals. However, Definition 7.14 on page 18 defines natural numbers to be those ordinal numbers such that they are either zero or every ordinal number less than it is a successor to some ordinal number. Thus 0 is an natural number, S(0) = 1 is an ordinal number, S(S(0)) = 2 is an ordinal number, and so forth. So without “finite” or “infinite” defined (See Definition 10.8 on page 28), Axiom 7 on page 19 assumes that there is set x which contains 0 and for every ordinal number in x its successor is also in x. Since any such set contains all natural numbers, the set of natural numbers exists by the Axiom of Comprehension. By Definition 10.8, the set of natural numbers and any set which satisfies Axiom 7 is infinite.
Where have you shown \(\forall x (x \in y )\) where the set y satisfies the axiom of infinity. Kunen provided such an argument. do more reading. it is clear you do not understand how to prove \(\forall x P(x)\) for some predicate P.
This requires more context, otherwise " \(\forall x (x \in y )\) where the set y satisfies the axiom of infinity" doesn't make any sense. As usual, chinglu, you are half-parroting mathematical statements without any understanding.
[Moderator: This thread has been moved and renamed to reflect chinglu's long struggle with the concept of infinity, Cantor's diagonal argument, formal mathematics, etc which all run afoul of the purpose of the main science forums and the established policies I was requested to police.] chinglu, you have not understood anything in Kunen. Autodidacts frequently mislearn subjects and abuse authorities and it appears you may fall into that category. Your abuse of people who are competent in the fields you profess interest in is ridiculously misguided. Please consult your PMs for a specific description of the policy. Also note that your abuse of posters (moderators or not) is completely off-base and cannot be tolerated. Be respectful and humble if you want to request answers to things that puzzle you. Continue to be arrogant and deceitful and expect to be judged harshly by both moderators and your peers.
well i explained to you that Kunen had a justification and you threw a temper tantrum. this also indicates you obviously do not understand the axiom. now, read this chapter 1 in his book regarding the axiom of infinity. please do not respond to me until you understand Kunen. http://www.cs.cornell.edu/courses/CS4860/2012fa/Kunen-SetTheoryBook-Cpht1.pdf
wow, just wow. If a set y satisfies the axiom of infinity then \(\forall x (x \in y )\) . Note since we are only dealing with finite ordinals as anyone should understand then x is a finite ordinal.
That does not follow. y could be the set of all subsets of the natural numbers having infinite cardinality. \( y = \left\{ z \; | \; z \in \mathcal{P}(\omega) \wedge \omega \preceq z \right\}\) This set exists because of the axiom of powersets and the axiom (schema) of comprehension. The correct way to write that is \( \vdash \forall x \in \omega ( x \in y ) \) or \(\vdash x \in \omega \to x \in y\) or \( \vdash \omega \subseteq y \) which isn't true. For example: \(\vdash y = \left\{ z \; | \; z \in \mathcal{P}(\omega) \wedge \omega \preceq z \right\} \to ( x \in \omega \to \neg x \in y )\).
"y could be the set of all subsets of the natural numbers" wrongo. we are assuming the axiom of infinity that only refers to 0 and successors. These are called natural numbers or finite ordinals. you need to put this thread back because of your error. the axiom does not refer to arbitrary subsets of natural numbers. then you need to admit your error. you also need to explain Kunen's justification for the axiom which you said needed none because axioms are assumed true. well Kunen provided one because Brouwer claimed it was not based on human intuition. now, use the link to the book i gave you and proceed. I'll provide links and such because of your lack of understanding. Solomon Feferman. “The development of programs for the foundations of mathematics in the first third of the 20th century” academia.edu. https://math.stanford.edu/~feferman/papers/foundations.pdf http://www.cs.cornell.edu/courses/CS4860/2012fa/Kunen-SetTheoryBook-Cpht1.pdf
Stumbling along all by yourself along the same old track eh chinglu? Same old approach, accuse everyone of being wrong when it's you who turns out to be wrong, or to have the wrong idea about the meaning of this or that mathematical fact. Not just once or twice, every time. It's worn out, it's kind of sad too. You think that coming up with this sort of thing , is underlining some kind of flaw; but the statement is flawed, infinity isn't a complete collection--you again underline instead, your own misunderstanding.
yeah, if only you had some kind of math proof then your post would be meaningful. but you don't have said proof do you. now, prove some kind of math proof or run along.
