Re: infinitely many nn's = infinite nn's?

george wrote:

On Mar 20, 10:35 pm, Phil <toob-head...@xxxxxxxxxxxxx> wrote:

How, exactly, would you prove that there are
infinitely many elements,
either in your set above, or in the set of natural numbers?


First, you would have to STATE SOME AXIOMS,
AND THEN, IN the context of THE SIGNATURE
(the alphabet, the language) underlying those axioms,
you would have to NOT "prove that there are infinitely"
many whatever, but rather: STATE what "infinity" MEANS!
AFTER you do that, GET BACK TO US!


I'm serious,


No, you're not, you're just stupid.

You, George Green, believe that there are infinitely many natural numbers, all of which are finite. Here, "infinitely many" is aleph-0, which is "actual infinity." Similarly, the "fact" that all the numbers are "finite" means that all numbers have values LESS THAN aleph-0, that the natural numbers are limited to "potentially infinite" values (magnitudes?).

But as for the axioms that lead to your belief that there are infinitely many (aleph-0) natural numbers, all of which are finite (< aleph-0), that's what I'm asking YOU! What are the AXIOMS used, in standard (versus nonstandard) mathematics, to either (1) STATE AS AN AXIOM that there are aleph-0 natural numbers, "infinitely many natural numbers," or (2), DEDUCE FROM those axioms that there are aleph-0 natural numbers? It damn sure isn't the "axiom of infinity" I found in Wikipedia and quoted, since that says NOTHING about whether the number of successors is actually or potentially infinite, i.e., aleph-0 or LESS THAN aleph-0.

I think that not even you believe that a number could have "infinitely many digits," aleph-0 digits -- or even infinitely many "marks," vertical lines like the Roman numerals (|, ||, |||, etc.) -- and have a FINITE (< aleph-0) value, but who knows? To be honest, it won't SURPRISE me one iota if you claim that there are some natural numbers with infinitely many digits, but that they, too, have only finite values. However, assuming that you believe that no numbers have infinitely many digits, then if you say that the axiom of infinity "does too" mean that there are aleph-0 numbers, fine, but what then stops this same axiom from stating that there are aleph-0 digits? Is the axiom of infinity MYSTICALLY LIMITED to the quantity of NUMBERS, meaning that one or more OTHER axioms must be used to determine the number of digits? What are those "other axioms?" If these are "easy questions that every SMART person knows" -- as opposed to "stupid" people like me -- then it should be no trouble at all for you or someone else to list the axioms in question. I'm not even asking you to explain the axioms or proofs, just LIST (1) the axioms that lead to aleph-0 numbers, and (2) the axioms that lead to < aleph-0 digits. Is that too much to ask of the honest, intelligent, open-minded, individuals that abound here?

Now, if you list the same set of axioms for both questions, I AM going to ask how those axioms led to different results, but again, for something as basic as the natural numbers, I'm sure such fundamental questions have standard answers, other than "they just mystically do that." Right?

Phil
.

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