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

In sci.logic, george
<greeneg@xxxxxxxxxx>
wrote
on 7 Apr 2007 11:46:42 -0700
<1175971602.064699.309100@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>:

On Apr 6, 11:50 am, Phil <toob-head...@xxxxxxxxxxxxx> wrote:
[re potential and actual infinity]
(2) that they are both
used in the definitions and axioms of the natural numbers,
but (3) WITHOUT being explicitly acknowledged!
That is a formula for certain disaster.

On Apr 6, 3:48 pm, "MoeBlee" <jazzm...@xxxxxxxxxxx> wrote:
Once you study some of the set theory books,

SHUT UP!
Jeezus, THAT is a formula for certain disaster (Phil trying
to study set theory?!? WHAT IS WRONG with you people??)

What is wrong with Phil studying set theory? As long as
he doesn't post here and bother us, I for one don't see
a problem... :-)

Of course it might confuse him further, but that's his problem. :-P


you'll see that your
predictions of disaster are unwarranted
and that the distinction you
feel to be so very crucial is not of any
import WITHIN such formal
theories as Z set theory.

You are completely missing the point.
You don't need to go ANYWHERE NEAR Z set theory
to investigate this issue. Even Peano Arithmetic is too far.
What you NEED is WEAK Peano Arithmetic:
0 is not a successor, successor is injective,
and induction. THAT IS ALL.

I'll agree but with one caveat: Phil wants to prove induction,
not assume it. I've yet to see his replacement axiom (given
what he's posted thus far, I'm not horribly hopeful).

The presence of
the induction schema legitimizes inductive (recursive)
definitions generally. That allows you to define the
"ancestral" of any function (viewed as a binary
relation; in fact, you can banish "pure" functors
from the signature altogether; a functor is just
a relation that behaves functionally, i.e., is just
its graph).

This is a concept quite unknown to Mathworld. Presumably,
if S is a set of ordered pairs such that no two
pairs have the same first element, one can define
ancestral_0(S) = S
ancestral_n(S) = ancestral_{n-1}(S)
union {(x,y): (x,z) is in S and (z,y) is in ancestral_{n-1}(S)
for some z in N}.

One can also describe this in terms of the concept of transitive
closure. This *is* known to Mathworld:

http://mathworld.wolfram.com/TransitiveClosure.html

and is probably a bit clearer than my explanation above.

The ancestral of successor is greater-
than, so you can define less-than as its converse.
Once you have defined less-than, you can define
addition.

Why would one want to go that route? The following
looks much more straightforward to me:

0 [axiom 1]
a + 0 =df a
0 + a =df a
a + succ(b) =df succ(a + b)
succ(a) + b =df succ(a + b)

and it turns out some of these are redundant.

Once one has defined addition on N, it's not too difficult to prove
that a + b > a and > b if a, b != 0.

Once you have defined addition, you can
define multiplication. THEN YOU ARE DONE.
THAT IS N, at 2nd-order anyway. All you need after
that is a definition of infinity. THIS is the point where
questions of what anyone might mean by potential
or actual infinity would LEGITIMATELY arise.

Well, infinity is a squirrely concept anyway. Aleph_null is much more
straightforward.

"lim(n -> oo) f(n) = oo" is a logical statement; neither
"oo" is really a number at all.

But somebody would've had to WALK Phil
through the above, STEP BY STEP, HOLDING HIS
HAND, BEFORE he could GET to this point, BEFORE
he could say anything about infinity.
Since I presented those axioms 4 times and Phil
never TOOK anyone's hand....



--
#191, ewill3@xxxxxxxxxxxxx
Useless C++ Programming Idea #889123:
std::vector<...> v; for(int i = 0; i < v.size(); i++) v.erase(v.begin() + i);

--
Posted via a free Usenet account from http://www.teranews.com

.

Relevant Pages

  • Re: Well Ordering the Reals
    ... most of the standard axioms would get scrapped ... one kind of set and infinity, and discrete counting systems form another, as in ... you claim that set theory is ...
    (sci.math)
  • Re: Can ZFC prove Addition is Associative?
    ... I've just looked it up in Halmos' "Naive Set Theory" (ad hoc ... This justifies my claim that "Induction is ... "The 'proof' of Induction is one axiom, the Axiom of Infinity. ...
    (sci.logic)
  • Re: Infinities and infinitesimals
    ... Wildberger lists the axioms of set theory and derides ... So really the question is "which axioms should you use for your set ... Here roughly are the main points: 1) An `infinity' is a growth rate of a ...
    (sci.math)
  • Re: infinitely many nns = infinite nns?
    ... These are axioms and assumed correct. ... state whether they should be used with potential, or actual, infinity. ... cannot use potential infinity for some conclusions, ... through Axiom, mathematical induction, or as it is sometimes called, ...
    (sci.logic)
  • Re: Well Ordering the Reals
    ... >>>Induction is stated axiomatically by Peano and used without regard to the ... then there must be some principle of set theory ... of the set of natural numbers and the axioms of set theory. ... > According to the inductive axiom, but the naturals are not *defined* to be ...
    (sci.math)
Loading