A number of set theories have been investigated which were obtained from ZF by restricting in various ways, or even deleting, some of the axioms of ZF-such as Power set, Aussonderung, Infinity, Replacement and Foundation. I define a sub-theory of ZF to be an axiomatizable theory formalized in the language of ZF, all of whose theorems are also theorems of ZF. Some well known sub-theories of ZF are not as weak as they appear to be. My question is: How weak can a sub-theory of ZF be and still be able to prove the existence of all the ordinal numbers that can be proved to exist in ZF? I would guess that Infinity and some version of Replacement would be required, but it is not clear to me what else might be needed.....My motive for asking this question comes from thinking about a result proved by A. Levy. He showed that cardinal numbers could not be defined in ZF if neither the Axiom of Choice nor the Axiom of Foundation were available. I wondered about how similar questions pertaining to ordinal numbers in ZF might be answered.
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asked Aug 12, 2014 at 18:47
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4$\begingroup$ What does it mean for the existence of an ordinal number $\alpha$ to be provable in ZF? If we have a definition of $\alpha$, say $\alpha$ is the unique ordinal satisfying $\phi(x)$, then provability of $\exists x\,\phi(x)$ makes sense. But what of undefinable $\alpha$? (Ignore them?) Worse, what if the same $\alpha$ has several definitions and ZF proves existence for one definition but not another? As a platonist, I understand "$\phi(x)$ defines $\alpha$" in terms of truth of $\phi$ in the real world; what can a non-platonist do? $\endgroup$– Andreas BlassAug 12, 2014 at 18:58
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2$\begingroup$ If I remember correctly, Lévy's theorem is not about defining an individual number (cardinal or ordinal) but rather about defining what it means to be a cardinal. $\endgroup$– Andreas BlassAug 12, 2014 at 19:00
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$\begingroup$ @Andreas: Levy showed that there existed no formula A(x) in the language of ZF, containing x as its one and only free variable, which expressed the statement "x is a Cardinal Number". Ordinal numbers are defined by such formulae in ZF and it is then proved in ZF that sets satisfying these definitions exist. I am asking whether all the axioms of ZF are needed to carry out these proofs. $\endgroup$– Garabed GulbenkianAug 13, 2014 at 19:33
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$\begingroup$ "Ordinal numbers are defined by such formulae" could mean (in analogy to what you wrote for cardinals) "there exists a formula ... containing x as its one and only free variable, which expressed the statement 'x is an ordinal number'." Or it could mean "for each ordinal $\alpha$, there exists a formula, containing x as its one and only free variable, expressing $x=\alpha$". The former is true; the latter is false. $\endgroup$– Andreas BlassAug 13, 2014 at 20:55
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$\begingroup$ Andreas, you are right. But what I am asking is, which of the axioms of ZF are actually needed to prove the existence of a all the sets which satisfy von Neumann's formula defining wjch sets are ordinal numbers and which belong to the set theoretic hierarchy of ZF. $\endgroup$– Garabed GulbenkianAug 14, 2014 at 21:02
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