Re: Am I a crank?

Tony Orlow wrote:
x=0;
while(finite(x))
{ add_to_list(x);
x++;
}

That's a loop generating the naturals.

If by 'loop' you mean 'eventually returning to the starting value',
then NO Peano structure is a loop.

The existence of a natural implying the existence of a next
natural is but an example of this kind of logical construction. Does
that seem like a wrong perspective to you?

In what context? Generally, I don't see existence of naturals as
arrived upon inductively. In first order PA, the natural numbers are
not mentioned in the theory itself. The natural numbers and the system
of them are a model of the theory, and each natural number is a member
of the universe of that model, but it is not the theory itself that
proves the existence of each natural number. Then, in set theory, any
given natural number can be proven to exist without recourse to
induction. We use induction to prove that certain PROPERTIES hold of
every natural number, but I think looking at existence itself as proven
that way is odd at best.

I don't see it as prof of existence as much as definition out of thin
air, which is fine for number systems.

An existence statement is either a theorem of the theory or it is not a
theorem of the theory. So when you say, "implying existence", we take
that as meaning that there exists a proof of existence (i.e., a proof
that there exists an object having the properties).

Secondly, I would like your opinion on inductive proof in the infinite
case.

How many times have I already posted to you that there IS transfinite
induction?

Many, but do you think that's what I'm talking about?

You're online "paper" starts out with a SWEEPING claim that in standard
mathematics there is no mathematical induction other than "in the
finite case" (or whatever particular wording you used). Your paper thus
announces the ignorance of its author right from the start.

I am aware that this concept is not compatible with transfinite
cardinalities or limit ordinals,

No, that's incorrect. I've been telling you that for months now. There
is transfinite induction.

Yes, but is that what I'm talking about? If it were allowable to prove
that 2x>x for all x>0, and omega or aleph_0 were considered greater than
0, then 2*aleph_0>aleph_0 would be a fact. However, it's not in the
standard construction.

Then give a logistic system and non-logical axioms in which you prove
it.
Otherwise, it's just a lot of hot air you're blowing.

Not EVERY property of finite sets is a propery of infinite sets. Just
to BEGIN with, the property of being FINITE is not going to be a
property of infinite sets. So, WHICH properties of finite sets must
also be properties of infinite sets? You claim that certain of these
cardinal arithmetic properties must carry from the finite to the
infinite. But it is arbitrary which ones they are. And it is NOT given
inductively, even by transfinite induction.

MoeBlee

.