# [FOM] Fwd: Re: Why ZF as foundation of mathematics?

Harry Deutsch hdeutsch at ilstu.edu
Mon Nov 11 18:36:02 EST 2013

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Subject: 	Re: [FOM] Why ZF as foundation of mathematics?
Date: 	Mon, 11 Nov 2013 09:09:42 -0600
From: 	Harry Deutsch <hdeutsch at ilstu.edu>
To: 	Zuhair Abdul Ghafoor Al-Johar <zaljohar at yahoo.com>, Foundations of
Mathematics <fom at cs.nyu.edu>

On 11/10/13 4:36 PM, Zuhair Abdul Ghafoor Al-Johar wrote:
> Dear Sirs,
>
> ZFC is been mentioned as the foundation theory of mathematics:
>
> Some quotes:
>
> "Today ZFC is the standard form of axiomatic set theory and as such is the most common foundation of mathematics" [Wikipedia]
>
> "Russell's Paradox can be avoided by a careful choice of construction principles, so that one has the expressive power needed for *usual mathematical arguments* while preventing the existence of paradoxical sets. See the supplement on Zermelo-Fraenkel Set Theory" [Thomas Jech: Stanford Encyclopedia of Philosophy]
>
> Also Thomas Jech in his "Introduction to Set Theory" makes similar remarks about ZF role in codifying mainstream mathematics.
>
> On the other hand it is well known that most of mainstream mathematics can be formalized in second order arithmetic, which is way weaker than ZF.
>
> So why ZF is mentioned for that purpose if much weak class\set theories can do the job of formalizing mainstream mathematics in set theory?
>
> I suggest a simple alternative class\set theory in mono-sorted first order logic language with a single extra-logical symbol, that of class membership, all objects are classes. The followings are the axioms:
>
> [1] Given a property there exists a class of all sets(i.e., elements of classes) that satisfy that property.
> [2] for any "singleton or empty" classes A,B; the class
> {x|set(x) & (x=A or x=B)} is a set.
> [3] Extensionality over all classes.
>
> This has the strength of second order arithmetic, has much simpler structure than ZF (proves only classes of hereditarily 'empty or singleton' sets and classes of pairs of those sets), and the axioms are dead simple.
>
> Best Regards,
>
> Zuhair Al-Johar
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It seems that you are using von Neumann's distinction between "elements"
and "non-elements" to avoid paradox.  Von Neumann's method is powerful
and neglected to some extent.  Fraenkel, Bar-Hillel and Levy
(Foundations of Set Theory, p.142) say this about this method (of
avoiding paradox): "His addition of proper classes to the universe of
set theory results from his discovery that it is not the existence of
certain classes that leads to the antinomies, but rather the assumption
of their elementhood, i.e., their being members of other classes.  In an
early edition of Foundations, these authors describe von Neumann's idea
as "daring." But they also note that it has the "peculiar" consequence
that while proper classes are "real objects, collections of proper
classes [even finite collections] do not exist."  Does this latter fact
have any effect on your version of second order arithmetic?"  Von
Neumann's method is very flexible.  I have used it recently to resolve
some long standing paradoxes of propositions, such as Russell's Appendix
B (of Principles of Mathematics) paradox and Kaplan's paradox.
("Resolution of some paradoxes of propositions" forthcoming in Analysis.)

Best Regards,

Harry Deutsch

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