Sentence examples for countable choice from inspiring English sources

In particular, no-one has refuted the prime ideal theorem or countable choice or DC.

For example one could add the principle of countable choice (AC0) or that of dependent choice (DC).

This seems to presuppose the truth of generalisations over all numbers and indeed countable choice, resources unavailable to a strict nominalist.

We recall that even if the full axiom of choice is not compatible with constructive and intuitionistic set theories, some consequences of AC, like countable choice, can be added to constructive and intuitionistic ZF (see the main document, section on Constructive Choice Principles).

(DC) is strictly weaker than (AC) but somewhat stronger than the Axiom of Countable Choice.

Direct attempts to extend the negative interpretation to analysis fail because the negative translation of the countable axiom of choice is not a theorem of intuitionistic analysis.

In the language of analysis, Markov's Principle and the negative translation of the countable axiom of choice are among the many non-intuitionistic principles which are function-realizable (by classical arguments) and hence consistent with FIM; cf. Kleene [1965], Vesley [1972] and Moschovakis [2003].

For this therapeutic situation, both the choice of radionuclide (with lower yield of countable photons emitted) and the activity concentration will be much different from the diagnostic situation.

This had to wait until 1963 when Paul Cohen showed that it is consistent with the standard axioms of set theory (which preclude the existence of atoms) to assume that a countable collection of pairs of sets of real numbers fails to have a choice function.

Since every countable level is itself countable (after all, there are only countably many possible defining formulas), and there are ω1 countable levels, there must be only ω1 real numbers.

Three-quarters have no countable assets.