Uniformity: Difference between revisions
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A similar justification is used for strongly compact cardinals. Given a \(\mathcal L_{\aleph_0,\aleph_0}\)-theory \(T\), if a \(\mathcal L_{\aleph_0,\aleph_0}\)-formula follows from \(T\), it follows from a finite subset of \\(T\\). The only infinite cardinals with this property are either \(\aleph_0\) or strongly compact cardinals, and if one wants to avoid the phenomenon stopping permanently after \(\aleph_0\), the existence of strongly compact cardinals must be assumed.
Barwise recounts about the above:<ref>J.
: The remarkable argument that strongly compact cardinals exist "by analogy with \(\omega\)" always reminds me of the goofang, described in ''The Book of Imaginary Beings'', by Jorge Luis Borges:
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