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I plan to write few foundations/set theory stubs including Skolem paradox. It will wait for a bit as the $n$Lab seems to be down at the moment.
The entry forcing has phrase downward Löwenheim-Skolem theorem. What does it mean downward in this phrase ? Is it a modifier at all ?
I created Skolem’s paradox.
The “downward” in the downward Loewenheim-Skolem(-Tarski) theorem refers to the fact that this is the version of the theorem which typically moves a model’s cardinality downwards; that is, it says that given a structure M in an infinite signature T, M has an elementary substructure of cardinality <= |T|. In particular, any theory T with a model (i.e., which is consistent) has a model of cardinality <= |T|. [This is the result which is given by a Skolem hull argument]
There is also an “upwards” Loewenheim-Skolem theorem, which, naturally, typically moves a model’s cardinality up; it says that given an infinite structure M in a signature T, for every cardinal k, M has an elementary extension of cardinality >= k. [This is the result which is given almost trivially by compactness]
By combining the two (typically: first using the upwards theorem to augment a structure to an elementary extension at least as large as desired, then nominally augmenting the signature to include as many distinct constants as necessary to maintain this cardinality lower bound, then using the downwards result to impose the same cardinality as an upper bound as well), one can of course get near-perfect control of the cardinality of models. But even in this combined form, using both results in conjunction, I think people still often say “downwards” or “upwards” when naming the theorem as appropriate to the final cardinality shift they intend to produce.
That is, you could combine the two into just one grand unified Loewenheim-Skolem theorem, of course, but the modifier serves to clarify the manner in which that one theorem is being applied (and, besides, the nature of the proof is such that there really are two rather separate results, given by rather separate arguments, which we might as well acknowledge).
Why don’t you start a more comprehensive version of Löwenheim-Skolem theorem in this manner ?
I started Gödel’s constructible universe.
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