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Single Idea 13675

[filed under theme 5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems ]

Full Idea

The Upward Löwenheim-Skolem theorem fails (trivially) with substitutional semantics. If there are only countably many terms of the language, then there are no uncountable substitution models.

Gist of Idea

Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails

Source

Stewart Shapiro (Foundations without Foundationalism [1991], 9.1.4)

Book Ref

Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.245

A Reaction

Better and better. See Idea 13674. Why postulate more objects than you can possibly name? I'm even suspicious of all real numbers, because you can't properly define them in finite terms. Shapiro objects that the uncountable can't be characterized.

Related Idea

Idea 13674 We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models [Shapiro]

The 24 ideas with the same theme [group of theorems about models involving infinities]:

17878 If a 1st-order proposition is satisfied, it is satisfied in a denumerably infinite domain [Skolem]
9913 The Löwenheim-Skolem Theorem is close to an antinomy in philosophy of language [Putnam]
10773 The Löwenheim-Skolem property is a limitation (e.g. can't say there are uncountably many reals) [Tharp]
10777 Skolem mistakenly inferred that Cantor's conceptions were illusory [Tharp]
13843 If it is a logic, the Löwenheim-Skolem theorem holds for it [Hacking]
10289 Up Löwenheim-Skolem: if infinite models, then arbitrarily large models [Hodges,W]
10288 Down Löwenheim-Skolem: if a countable language has a consistent theory, that has a countable model [Hodges,W]
17813 Löwenheim-Skolem says any theory with a true interpretation has a model in the natural numbers [White,NP]
17790 No Löwenheim-Skolem logic can axiomatise real analysis [Mayberry]
13648 The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro]
13659 Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes [Shapiro]
13658 Downward Löwenheim-Skolem: each satisfiable countable set always has countable models [Shapiro]
13675 Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails [Shapiro]
10292 Downward Löwenheim-Skolem: if there's an infinite model, there is a countable model [Shapiro]
10590 Up Löwenheim-Skolem: if natural numbers satisfy wffs, then an infinite domain satisfies them [Shapiro]
10296 The Löwenheim-Skolem Theorems fail for second-order languages with standard semantics [Shapiro]
10297 The Löwenheim-Skolem theorem seems to be a defect of first-order logic [Shapiro]
10234 Any theory with an infinite model has a model of every infinite cardinality [Shapiro]
10988 Any first-order theory of sets is inadequate [Read]
10160 Löwenheim-Skolem says if the sentences are countable, so is the model [Feferman/Feferman]
10159 Löwenheim-Skolem Theorem, and Gödel's completeness of first-order logic, the earliest model theory [Feferman/Feferman]
13539 The LST Theorem is a serious limitation of first-order logic [Wolf,RS]
17748 The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki]
17929 Löwenheim proved his result for a first-order sentence, and Skolem generalised it [Colyvan]