The formula alone says that there are at least elements in the universe of the structure. In other words, for .

Consider the set and use the form of compactness theorem from this thread.

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- Nov 14th 2011, 01:01 AM #1

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## Fol 10

Enderton 2.6.8

Assume that is true in all infinite models of a theory . Show that there is a finite number such that is true in all models of for which has or more elements.

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The sentence for there are at least things for is

, where

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Any hint to get started?

Thanks.

- Nov 14th 2011, 01:45 AM #2

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## Re: Fol 10

The formula alone says that there are at least elements in the universe of the structure. In other words, for .

Consider the set and use the form of compactness theorem from this thread.

- Nov 14th 2011, 04:41 AM #3

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## Re: Fol 10

- Nov 14th 2011, 06:57 AM #4

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- Nov 15th 2011, 01:03 AM #5

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## Re: Fol 10

- Nov 15th 2011, 01:34 AM #6

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- Nov 15th 2011, 01:53 AM #7

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## Re: Fol 10

My understanding is,

is true in all infinite models of a theory T by assumption.

By compactness theorem, is true in some finite models of a theory T having cardinality k.

However, I think we still have to establish that is true in all finite models of a theory T having cardinality greater than k.

Is this idea wrong?

- Nov 15th 2011, 01:57 AM #8

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