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

"Phil" <toob-headman@xxxxxxxxxxxxx> wrote in message
news:463CF9A1.8030903@xxxxxxxxxxxxxxxx
And not only did I NOT believe that it is an INHERENT
property of ANY infinite set of natural numbers that begins with 0 or 1
and skips no numbers, that if n is in N, then so is 2n, I STILL don't
believe it!

Why not? Is "skips no numbers" not equivalent to the axiom of succession
and the axiom of induction?

EVERY "proof" that I have seen that claims to prove that
this is an inherent property of infinite sets simply ADDS this belief as
a new axiom, period. EVERY ONE!

They are all based on an axiom: the successor of a natural number is also a
natural number, and a definition of addition that is in terms of the axiom
of succession through a recursion. You need to claim that your definition
of N through your properties, above, is isomorphic to the Peano Axioms.
Perhaps your properties do not admit addition at all?

Do you think that if n is in N, then so
is 2n? That's an axiom; prove otherwise Jesse! PROVE, from the simple
fact that you have a set N1 beginning with 0 or 1, that skips no
numbers, with INFINITELY many elements, that there is no such thing as a
set N2 that (1) has twice as many elements as N1, (2) has every element
in N1, and (3) has infinitely many elements that are NOT in N1, and do
so WITHOUT simply adding that belief, or a LOGICAL EQUIVALENT of that
belief, as an axiom. Go ahead. Because you will, I assure you, be the
FIRST person in history to do so. Nobody else has EVER done that!

Using your properties, perhaps. They do not appear to be logically
equivalent to the properties of N.

Instead, let's talk about some simpler questions.

(1) Do you agree that whenever n is in N, then so is n+1?

(2) And hence, whenever n and m are in N, then so is n + m?

(3) And do you agree that 2n = n + n?

Which of these are not obvious?

Jesse, "obvious" doesn't cut it when dealing with infinite sets. Which
is more obvious, that eliminating half a set's elements reduces its
size, or that it leaves its size untouched? Yes, regardless of which
option we choose, when dealing with infinite sets, we are going to have
SOME results that appear to be contradictory. However, you CANNOT, as a
true mathematician, arbitrarily say that "Well, THIS set of apparently
contradictory features is just fine, but THAT set is not." If axiom P
leads to a self-consistent -- but not OBVIOUSLY so -- branch of
mathematics, and so does not-P, then they are BOTH viable options, true?

If you're giving me a choice between having a reliable defintion of
addition, and a system without one, guess which one I'll pick!


karl m


.

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