Uniformity

Uniformity is a justification used for existence of some very large large cardinals, such as measurable and strongly compact cardinals. As there is no current characterization of measurable cardinals by closure under certain operations as there are with some smaller large cardinals, uniformity is often the method of choice for justifying their existence.

The principle behind the uniformity justification states that the set-theoretic universe should remain interesting when progressing through its ranks, and that the alternative would mean "as if the universe had lost its complexity at the higher levels, as if it had flattened out, become homogeneous and boring." Since there is a nontrivial \(\aleph_0\)-additive two-valued measure on the subsets of \\(aleph_0\\), it would be expected that this property holds for larger cardinals as well, otherwise this phenomenon would occur only for \(\aleph_0\) and then never reoccur for any larger cardinals.

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:

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 yarns and tall tales of the lumber camps of Wisconsin and Minnesota include some singular creatures, in which, surely, no one ever believed...

There's another fish, the Goofang, that swims backward to keep the water out of its eyes. It's described as "about the size of a sunfish, only much bigger".