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I sometimes come across the phrase, "a standard model of ZFC." Is this a rigorous concept? If so, how does one define it?

goblin GONE
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1 Answers1

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A standard model of ZFC is a pair $(M,E)$ such that $M$ is non-empty and $E$ is well-founded, and $(M,E)\models\sf ZFC$. Using Mostowski's collapse lemma this is isomorphic to a transitive set, $N$ such that $(N,\in\upharpoonright_N)\models\sf ZFC$.

Standard models are exactly those models of ZFC whose ordinals are well-ordered (externally), and if they are transitive then their ordinals are exactly some ordinal $\alpha$.

Asaf Karagila
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  • Removed the "the". – goblin GONE Feb 22 '13 at 01:35
  • So essentially, you pick an ambient set theory and then you define that a standard model of ZFC is a pair of sets $(M,E)$ (in the sense of the ambient theory) such that $E$ is well-founded according to the ambient theory, and $(M,E)\models \mathrm{ZFC}$? – goblin GONE Feb 22 '13 at 01:37
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    @user18921 That would be any model. We also require that $E$ is well-founded. – Trevor Wilson Feb 22 '13 at 01:38
  • @user18921: Close. We require $E$ to be well-founded. If the universe satisfies ZF then we can collapse this to another set $N$ and replace $E$ by the true $\in$ of the background universe. – Asaf Karagila Feb 22 '13 at 01:39
  • Yes I fixed the comment. – goblin GONE Feb 22 '13 at 01:39
  • Asaf, does this use of the word "standard" have any connection to the "standard" semantics of second-order logic? I was thinking that the word "standard" often seems to mean: "Not only does it look like a model from inside; but also from the outside." – goblin GONE May 13 '13 at 14:55
  • @user18921: Standard semantics of second-order logic? – Asaf Karagila May 13 '13 at 14:58
  • Umm yes I don't know if I'm using the correct terminology. Standard as opposed to the Henkin semantics. – goblin GONE May 13 '13 at 14:59
  • @user18921: You mean full semantics. There is no standard semantics to second-order logic. This is why there are two common of them. – Asaf Karagila May 13 '13 at 15:00
  • Oh. I was reading the paper "SECOND-ORDER LOGIC, OR: HOW I LEARNED TO STOP WORRYING AND LOVE THE INCOMPLETENESS THEOREMS" and it uses the term standard rather than full. Anyway yes that is what I meant. Is there any connection between them? – goblin GONE May 13 '13 at 15:02
  • @user18921: Not that I am aware of. I think that the term standard indicates that all the ordinals are real ordinals, which is true if and only if the model is well-founded (and transitive, but we can always collapse it to a transitive model and make $E$ to be $\in$). – Asaf Karagila May 13 '13 at 15:15
  • Cool, thanks. Have a great day. – goblin GONE May 13 '13 at 15:25
  • Hey one more question - does working with standard models of ZFC ensure that the $(\mathbb{N},0,S)$ of the model will be isomorphic with the $(\mathbb{N},0,S)$ of the larger universe? – goblin GONE May 19 '13 at 02:57
  • @user18921: Yes. You can prove by induction that every well-founded model of the same theory is equal to $\Bbb N$ (in a given model). So inside a universe we can prove that standard models have standard integers, and no more. – Asaf Karagila May 19 '13 at 03:10
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    How can a well-founded model of ZFC be equal to $\mathbb{N}$? – goblin GONE May 19 '13 at 03:11
  • What? No. I meant the theory of the integers with zero and successor. – Asaf Karagila May 19 '13 at 03:24