Re: Well Ordering the Reals

David R Tribble said:
Tony Orlow wrote:
1. 0<x<=1 -> finite(x)
2. finite(x) -> finite(0-x)
3. finite(x) -> finite(1/x)
4. finite(x) -> finite(2^x)


David R Tribble said:
The rules say nothing about "non-finite" numbers, however, so it's
impossible to see how they show that Peano's 5th axiom implies the
existence of infinite naturals.


Tony Orlow wrote:
I believe I had added:
5. x<>0 and not(finite(x)) -> infinite(x)


Tony Orlow wrote:
David R Tribble said:
Which, given the definitions above, looks like an empty set.


David R Tribble said:
Rule 1 says that all x in (0,1] are finite.
Rule 2 says all x<0 are finite.
Rule 3 says all x>1 are finite.
Rule 5 says that 0 is not finite nor infinite.

So there are no values of x derivable from your rules that
satisfy infinite(x).


Tony Orlow wrote:
Not if you can't properly read the rules.

It's a waste of time, but let's try again:

Rule 1 say that all x in (0,1] are finite.
(Maybe - you don't define x, so we can't be sure about this.)
Correct

Rules 1 and 3 define all x>0 to be finite.
Only finite x. Since 0 is not included as finite, 1/0 is also not included as
finite, but the reciprocal of any finite is finite.

Rules 1, 2, and 3 define all x<>0 to be finite.

Rule 2 says that if x is finite, then the negative of x is finite, so 1, 2 and
3 together include (0,1], [-1,0), and [1,oo) and (-oo,-1]


Rule 5 says that 0 is neither finite nor infinite.

0 was never included as finite. What five says is that if x is not zero or
finite, then it is infinite.


That exhausts all the reals, so there is no x derivable from your
rules that satisfies infinite(x).

No finite reals satisfy infinite(x). The inverses of 0 and the infinitesimals
(which are not included in these rules so far, since we were talking finite vs.
infinite), are infinite.



Tony Orlow wrote:
Anything that is not zero or finite is infinite, whether positive or negative.

So where do these "anythings" come from? I don't see anything
in your rules that define anything but the finite reals and zero.

From the inverse of 0 and the infinitesimals. Shall I add more axioms?

--
Smiles,

Tony
.

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