Uniformity: Difference between revisions
Jump to navigation
Jump to search
Content added Content deleted
No edit summary |
No edit summary |
||
| Line 5: | Line 5: | ||
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. |
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. |
Barwise recounts about the above:<ref>J. Barwise, ''Admissible Sets and Structures'', p.364. Perspectives in Mathematical Logic (1975).</ref> |
||
: 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: |
: 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: |
||