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In Model theory, the definition of a model is a set. Can it be a proper class?

ZFC has a model and maybe some models is a proper class. Definition of a model needs to include a proper class.

Is it correct? Id like to ask your opinion.

tarou
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3 Answers3

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A model is, by definition, a set. So no, "class models" are not models in the usual sense. Lots of theories besides ZFC also have "class models," but that's a different sort of thing - and really can't be nicely studied without working in a class theory like NBG or MK.

Noah Schweber
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  • thank you for your comment. I understand that usually model is a set . it can be proper class but it can not be formalized in ZFC. – tarou Mar 31 '16 at 01:42
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    Tent and Ziegler's book "A Course in Model Theory" has a nice exposition of class sized models of a theory in the setting of NBG. – greg Mar 31 '16 at 22:02
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Normally a model is understood to be a set, but it is not rare for people to consider proper class models. Sometimes they do this informally with no rigorous justification. One way to formalize it is in Godel-Bernays set theory where proper classes actually exist.

ZFC has models if it is consistent (and pretty much everyone who cares believes that ZFC is consistent). And, because ZFC is a first order theory, if it is consistent then it has set models, indeed it has countable models. People do sometimes consider proper class sized models of ZFC too -- which can be formalized in various ways. One way is to use Godel Bernays, of course. A more common way is to use "definable classes," as described in difference between class, set , family and collection and at a higher level at difference between class, set , family and collection

Community
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Colin McLarty
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Assuming the existence of a class-size model of a theory is very common. Sometimes model theorists want to talk about how "large" saturated models work over a "small" set of parameters. There a formal notions that define "large" and "small" when talking about set sized models (e.g. Hodges's $\kappa-$bigness). However, sometimes these details clutter the big picture. So, it isn't uncommon for people to assume that there exists a class size model $\mathbb{M}$ (often referred to as a monster model - but this term can also just mean "large enough") where "large" means "a class sized subset of $\mathbb{M}$" and "small" means "a set sized subset of $\mathbb{M}$".

Kyle Gannon
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