TY - JOUR
T1 - On Löwenheim-Skolem-Tarski numbers for extensions of first order logic
AU - Magidor, Menachem
AU - Väänänen, Jouko
PY - 2011/6
Y1 - 2011/6
N2 - We show that, assuming the consistency of a supercompact cardinal, the first (weakly) inaccessible cardinal can satisfy a strong form of a Löwenheim-Skolem-Tarski theorem for the equicardinality logic L(I), a logic introduced in [5] strictly between first order logic and second order logic. On the other hand we show that in the light of present day inner model technology, nothing short of a supercompact cardinal suffices for this result. In particular, we show that the Löwenheim-Skolem-Tarski theorem for the equicardinality logic at κ implies the Singular Cardinals Hypothesis above κ as well as Projective Determinacy.
AB - We show that, assuming the consistency of a supercompact cardinal, the first (weakly) inaccessible cardinal can satisfy a strong form of a Löwenheim-Skolem-Tarski theorem for the equicardinality logic L(I), a logic introduced in [5] strictly between first order logic and second order logic. On the other hand we show that in the light of present day inner model technology, nothing short of a supercompact cardinal suffices for this result. In particular, we show that the Löwenheim-Skolem-Tarski theorem for the equicardinality logic at κ implies the Singular Cardinals Hypothesis above κ as well as Projective Determinacy.
KW - equicardinality quantifier
KW - Härtig-quantifier
KW - Löwenheim-Skolem theorem
KW - supercompact cardinal
UR - http://www.scopus.com/inward/record.url?scp=80052766923&partnerID=8YFLogxK
U2 - 10.1142/S0219061311001018
DO - 10.1142/S0219061311001018
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AN - SCOPUS:80052766923
SN - 0219-0613
VL - 11
SP - 87
EP - 113
JO - Journal of Mathematical Logic
JF - Journal of Mathematical Logic
IS - 1
ER -