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

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


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.


What axiom of succession? Do you mean the axiom that, if n is a
natural number, then so is n + 1?

No, the axiom that states that if n exists, so does n + 1. What you wrote above is an example of induction, not succession. How do you "know" that infinitely many numbers exist? That's the successor axiom. But again, if you use ONLY that axiom -- well, combined with the option of "succeeding," of adding 1 to the current number, infinitely many times -- then you have to conclude not only that there are infinitely many numbers, but also that there are infinitely large numbers, due to what I said below, namely that if you add 1 to 0 infinitely many times, you have (1) infinitely many additions, corresponding to infinitely many numbers, and (2) infinitely large numbers, since you either added an infinite number to 0, or multiplied 1 by an infinite number. The axiom of induction is what LIMITS the naturals to finite values, but what everyone overlooks is that you cannot ALSO use this axiom one minute, but then fail to use it the next. The naturals are a COMPOSITE of their axioms. In most cases, those axioms are "additive," but here and there, they limit one another. Induction limits BOTH the number of naturals (if n is preceded by finitely many naturals, then so is n+1, for ALL n in N), and also the SIZE of the naturals, as you noted above (if n is finite, then so is n+1, for ALL n in N). That doesn't mean, as George often says, that although no number is preceded by infinitely many numbers, that the SET is somehow "mystically" infinite. When he strays too long from induction, you can catch him saying that "there are infinitely many naturals," but by induction, this REALLY IS impossible. After all, if you can IGNORE induction when it comes to the number of naturals, then why can't you ignore induction when it comes to the size of the naturals?

Phil

That axiom does *not* prove there are infinitely large numbers.

If you think otherwise, please give a clear proof of the theorem
from the axiom that each natural number has a successor which is also
a natural number. When we see the steps involved in the argument,
perhaps we can tell whether the claim actually follows from the axiom.


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.


There is no axiom which says anything about adding 1 infinitely many
times. Sorry.

And there is no theorem that there is a natural number which is the
result of adding one to itself infinitely many times, either.


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.


No one ever claimed that if you multiply a natural number by an
infinite ordinal, you get a finite result.


[...]

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