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As part of a homework assignment for a logic class, I'm supposed to find a finite set $\Gamma$ (I believe of wffs) such that any model of $\Gamma$ has an infinite domain. This is for the predicate calculus, and I'm really not sure how to get started. Every solution I can think of seems to depend on the interpretation of a predicate relation. For example, since infinite sets have no minimal or maximal element I could do something like

$$(\forall x_1) \neg A^{2}_{1}(x_1, f^{2}_{1}(x_1))$$

where $A^{2}_{1}$ is equality and $f^{2}_{1}$ is the successor function. However, this involves a specific interpretation and my understanding is that the set $\Gamma$ should not have any associated interpretation. I could really just use a hint to get me started.

Danny
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    Have a look to the question http://math.stackexchange.com/questions/1627165/does-a-finite-first-order-theory-which-has-a-model-always-have-a-finite-model/1632532#1632532 (Alex Kruckman gave a great answer to it). – Graffitics Apr 03 '16 at 09:35

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HINT: You can actually use basically the same idea. All you need is a binary relation symbol $R$ and axioms saying that the relation is a strict linear order, together with $\forall x\,\exists y\,R(x,y)$.

Brian M. Scott
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  • Is it possible to do without introducing axioms? Based on the framing of the assignment and the general noncomplexity of the class I'm taking, I don't think the intention is for us to use additional axioms. I think we're just supposed to stick with the five generic ones given in Mendelson. – Danny Apr 03 '16 at 03:34
  • @Danny: The axioms are the members of the set $\Gamma$ that you’re to produce. – Brian M. Scott Apr 03 '16 at 03:35
  • @Danny: That last comment may not have been entirely clear. What I’m calling axioms are simply the formulas that are to be included in $\Gamma$. The term axiom has no special significance here: all I meant by it was the formulas that $R$ is required to satisfy. – Brian M. Scott Apr 03 '16 at 03:43
  • Oh ok, I see. Thank you! That hint helps. – Danny Apr 03 '16 at 03:45
  • @Danny: You’re welcome! – Brian M. Scott Apr 03 '16 at 03:46