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Suppose that Replacement holds only for parameter-free formulas. I need to Prove that the full schema of Replacement holds.

the full schema of Replacement states: If $ϕ(u,v,p_1,...,p_n)$ is a formula in the language of set theory, for every choice of parameters $p_1,...,p_n$, if for some x we can prove that for u ∈ x there exists exactly one v such that $ϕ(u,v,p_1,...,p_n)$ holds, namely that $ϕ$ defines a function on $x$, then there is $y$ which is the range of this function;

$${\forall p_1,\ldots,p_n}~{{\Big(\forall x~\big((\forall u {\in} x)(∃!v~ϕ(u,v,p_1,...,p_n))\big)}\to{\big(\exists y~\forall v~(v{\in}y \gets\hspace{-2ex}\to(\exists u {\in} x)~ϕ(u,v,p_1,...,p_n))\big)\Big)}}$$

I have three questions:

  1. who are , $p_1,...,p_n$? are they variables of the language, or are they sets? and how are they different from $u$ and $v$?
  2. can I get an example of a formula ϕ(u,v,p1,...,pn) please?
  3. how can the statement be proven?
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  • $\begingroup$ This is not a trivial theorem at all. Do you know the analogous theorem for PA, that you can use only parameter-free induction axioms? The proofs are a bit similar, but I think the PA one is a bit easier to grasp, since it is easier to understand the natural numbers. $\endgroup$ – Asaf Karagila Aug 25 '17 at 10:23
  • $\begingroup$ Also, if you don't know yet that parameters are variables, and that all objects in the case of ZFC are sets, then perhaps you're not ready for this sort of theorem. $\endgroup$ – Asaf Karagila Aug 25 '17 at 10:24
  • $\begingroup$ When I studied logic we never used the term "parameter", only "variables". and since this formula is written in a first order language, I assumed that ,p1,...,pn are variables. I'll ask it differently: $\endgroup$ – Mike. R. Aug 25 '17 at 11:08
  • $\begingroup$ I'll ask it differently: 1. these axioms are written in a specific first order language we call the language of set theory .thus, u, v, etc. are variables of the language. if we could find a collection of objects that satisfies these axioms, namely - if we can find a model for our axioms, then the objects of this structure are called "sets". is that right? 2. when it says " parameter-free formulas", dos that mean with out p1,...,pn, or with out u,v,p1,...,pn? I dont know the analogous theorem for PA. $\endgroup$ – Mike. R. Aug 25 '17 at 11:24
  • $\begingroup$ (1) Yes. (2) Yes. It means that the formula has a single free variable, so it is $\phi(u)$, and not $\phi(u,p_1,\ldots,p_n)$. (3) The first-order axiomatization of Peano Arithmetic has an induction schema, which also utilizes parameters, and there too you have a theorem that parameter-free induction is enough. $\endgroup$ – Asaf Karagila Aug 25 '17 at 11:36
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A comment by Goldstern to a related MO question provides a link to a beautiful note by Schindler and Schlicht that contains all the details. The idea of the proof is to encode all parameters into $x$ using ordered pairs. These are later decoded by a suitably modified formula $\phi$. As it is written in the note, result itself is due to Levy.

I believe that questions 1 and 2 were sufficiently addressed in the comments by Asaf Karagila.

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