Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)

From: Daniel Grubb (grubb_at_lola.math.niu.edu)
Date: 11/03/04 

Date: 3 Nov 2004 16:45:05 GMT

>Surely that's a good idea. Here come some:

>http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html

This one attempts to use quantifiers, but fails
miserably. I wouldn't rely on it.

>http://planetmath.org/encyclopedia/ZermeloFraenkelAxioms.html

This one has problems with notation. For example, it writes P(x)
when it should be P(X). Both regularity and the axiom of infinity
are stated in non-standard ways. It isn't clear to me that this
version of regularity is equivalent to the usual one.

>http://www.brainyencyclopedia.com/encyclopedia/a/ax/axiomatic_set_theory_1.html

This one is better. The main consideration I have is that there is no
definition of the term "proposition". This is typically done recursively.
There is also some ambiguity due to the lack of logical notation
(which is what the first one was trying to do).

For example, the axiom of the empty set states that there is a set
with no elements. It is perhaps better to say something like

"There exists an x such that for all y, not(y is an element of x)"

The point is that "y is an element of x" is an undefined term,
so should be the thing used in the axioms.

Similarly, in the axiom of infinity, it is probably better
to not explicitly use {} and union, but write out the axiom
in terms of primitive terms.

Not too bad overall, though.

>http://www.fact-index.com/z/ze/zermelo_fraenkel_set_theory.html

This one is essentially the same as that from the previous link.
The only significant difference is the axiom of choice, where the
product of non-empty sets is not defined. If it were properly defined,
this version wopuld be equivalent to the previous version.

>http://www.webster-dictionary.org/definition/Zermelo-Fraenkel%20set%20theory

The main difference I see for this one is the subset axiom. This follows
from the axiom of replacement, so this set is equivalent to the
previous two. The statement of the axiom of choice is clunky, though.
I prefer the statement from the link two back from here.

>Do all these pages represent ZFC correctly? Can I rely on this?

I would have the recursive definition of a proposition as part of the
axiom system. The links that had correct statements of ZFC did not do
this. They also used terms that were not defined at the time they
were used. This is bad, IMHO. I'd say that you should stick with the last
3 of your links.

--Dan Grubb

Relevant Pages

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