Re: Cantor's circular "proof" that evens = integers

Jesse F. Hughes wrote:
Phil <toob-headman@xxxxxxxxxxxxx> writes:


Notice that instead of using the entire set of axioms to determine how
many numbers exist, you used just one axiom. Since the characteristics
of the natural numbers are determined by the ENTIRE set of axioms, you
CANNOT use just ONE axiom to determine some characteristic; that is an
error of logic and reasoning.


Sorry, Phil, but this is just nonsense. If we can prove that
something follows from a single axiom, then it follows from a set of
axioms that include the single axiom.

Then by the axiom of succession, since there is no end to the numbers that can be produced, there are infinitely large numbers. And remember, by your own rules, you CANNOT use another axiom to contradict this.

Honestly, IF you add 1 to an accumulator INFINITELY many times you WILL end up with an infinite number, since adding 1 infinitely many times -- lets assume that "infinitely many" = omega -- really is the same thing as adding omega, and 0 + omega = omega, an INFINITE number.

Or, you could use multiplication. As several here proved to me years ago (I THINK George was one of them), multiplication is NOT merely repeated addition, but adding 1 w times, where w = any infinite number you like, nonstandard or otherwise, IS equivalent to "simple" multiplication, where we multiply 1 times the infinite number w, which I suspect even YOU will have to admit ends up equal to the infinite number w.

Face it, given succession, the ONLY reason we do NOT have infinitely large numbers is because ANOTHER axiom, namely induction, IS ALSO USED. And if you HAVE to use BOTH succession AND induction when you conclude how LARGE the naturals numbers can be, WHAT makes you think that you do not have to do the same when you conclude how MANY natural numbers exist? Do you think that if induction doesn't OBVIOUSLY have to be used, then it DEFINITELY doesn't have to be used? I hope you're not this sloppy with all your proofs!

Did my diagram showing both the accumulator and the resulting sets of marks representing the numbers really mean NOTHING to you? You DID NOTICE, I trust, that the accumulator -- which represents at any moment the NUMBER of naturals we have generated -- CANNOT become infinite without also making an infinite NUMBER! SO, if your AXIOMS -- you know, PLURAL, as in MORE THAN ONE axiom -- limit the naturals to FINITE values, then there CANNOT be infinitely many of them!!! It's either that, or you think that although the accumulator now has infinitely many marks in it, and continues to add one mark per number, that the resulting numbers are no longer equal to the accumulator.

Jesse, instead of just assuming that mathematicians have created a perfectly consistent and logical structure, THINK FOR YOURSELF for a moment, and ask yourself questions ABOUT THE STRUCTURE that results FROM the current beliefs. By definition, the ONLY reason that the accumulator starts having infinitely many marks is because it has RUN OUT OF finitely many marks. It "shifts" to infinitely many marks ONLY because, and ONLY after, there are no more sets of FINITELY many marks to make. If you ASSUME that you have indeed added so many marks to an accumulator that it now has infinitely many marks w, where w again can be ANY infinite number, no matter how small -- and if you do NOT assume this, then there CANNOT be infinitely many numbers -- then you MUST have created every possible set of finitely many marks that it is possible to create! There are no more! And with each additional mark, you MUST be creating another INFINITE number.

The reason I say that we must look at the structure that RESULTS from the current beliefs, instead of simply checking to make sure that each belief does indeed follow from valid premises -- and I will be the first to admit that each of the current beliefs concerning the naturals DOES necessarily and properly follow from valid premises -- is because a direct examination of the resulting structure can sometimes show flaws that no one could otherwise find. It's a validation technique, and a very good one at that. IF no errors were made, THEN the resulting structure, as well as the individual proofs, will have no flaw. But in the case of the natural numbers, the current beliefs produce a structure that is VERY flawed, and simply verifying that each individual conclusion is valid CANNOT reveal those flaws!

At the very least, please tell me that you think we should at least CHECK to see if maybe, just maybe, we have to use more than one axiom when deducing how MANY naturals exist within the set of finite naturals. As I think I said to you in another post, if necessary, LIE and tell me you think we MIGHT have to use more than one axiom when deducing some of the facts concerning the natural numbers.

Phil
.

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