Talk:Weakly compact cardinal: Difference between revisions
Latest comment: 10 months ago by C7X in topic "A relatively convoluted definition"
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Then there is only one major change between the compactness theorem and the weak compactness property: |
Then there is only one major change between the compactness theorem and the weak compactness property: |
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* When \(\Gamma\) is a set of \(\mathcal L_{\omega,\omega}\)-sentences <u>of size \(<\aleph_0\)</u>, if every subset of \(\Gamma\) of size \(<\aleph_0\) has a model, then \(\Gamma\) has a model. |
* When \(\Gamma\) is a set of \(\mathcal L_{\omega,\omega}\)-sentences <u>of size \(\underline{<\aleph_0}\)</u>, if every subset of \(\Gamma\) of size \(<\aleph_0\) has a model, then \(\Gamma\) has a model. |
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* When \(\Gamma\) is a set of \(\mathcal L_{\kappa,\kappa}\)-sentences of size \(<\kappa\), if every subset of \(\Gamma\) of size \(<\kappa\) has a model, then \(\Gamma\) has a model. |
* When \(\Gamma\) is a set of \(\mathcal L_{\kappa,\kappa}\)-sentences of size \(<\kappa\), if every subset of \(\Gamma\) of size \(<\kappa\) has a model, then \(\Gamma\) has a model. |
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where the change is the |
where the change is the underlined part. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 21:15, 29 August 2023 (UTC) |
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Revision as of 21:17, 29 August 2023
"A relatively convoluted definition"
In my opinion as I've been working more with infinitary languages, it seems like the compactness property of weakly compact cardinals is not very complicated. The compactness theorem already admits some extensions, with the most attention directed to what to replace the word "finite" in the compactness theorem with.
Then there is only one major change between the compactness theorem and the weak compactness property:
- When \(\Gamma\) is a set of \(\mathcal L_{\omega,\omega}\)-sentences of size \(\underline{<\aleph_0}\), if every subset of \(\Gamma\) of size \(<\aleph_0\) has a model, then \(\Gamma\) has a model.
- When \(\Gamma\) is a set of \(\mathcal L_{\kappa,\kappa}\)-sentences of size \(<\kappa\), if every subset of \(\Gamma\) of size \(<\kappa\) has a model, then \(\Gamma\) has a model.
where the change is the underlined part. C7X (talk) 21:15, 29 August 2023 (UTC)