For the Thing Theorist “to be is to be countable.”
Second, if it has countable models, it has models of any higher cardinality.
Nick Mitchell's Northern Fail app took Northern Rail’s countless failures and made them countable (WIRED).
Nothing is countable unless there exists a counter.
Truth definition in L(ω1,ω) for a countable base language L.
When L is countable, the situation is a little more involved.
For infinite languages, they assumed an axiom of countable additivity for probabilities.
On the assumption that T has a model, the Löwenheim-Skolem theorems ensure that it has a countable model.
But it may still leave an obvious question unanswered: how can a countable model of ZFC satisfy such a formula?
S = ⟨U, F⟩ is a probabilistic space, with U countable and where F is a Boolean sub-algebra of the power set of U.
Let A be a countable transitive set such that LA is a sublanguage of L(ω1, ω) and let Δ be a set of sentences of LA.
It implies the axiom of choice for countable sets of sets but is incompatible with the unrestricted axiom of choice.
Veblen in 1908 extended the initial segment of the countable for which fundamental sequences can be given effectively.
The Löwenheim-Skolem theorem says that if a first-order theory has infinite models, then it has models whose domains are only countable.
Its syntax is based upon two sets of symbols: a countable set Φ0 of atomic formulas and a countable set Π0 of atomic programs.
The effect of abandoning even countable choice is the exclusion of many theorems that, as they stand, are proved using sequential, choice-based arguments.
Kolmogorov comments that infinite probability spaces are idealized models of real random processes, and that he limits himself arbitrarily to only those models that satisfy countable additivity.
The point Norton makes about the impossibility of a uniform distribution over a countable set of time intervals holds also for the time at which we might expect the space invader to occur in the second type of case.
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.
As for set theory, the failure of categoricity was already taken note of by Skolem in 1923, because it follows from the Löwenheim-Skolem Theorem (which Skolem arrived at that year; see Skolem 1923, based on Löwenheim 1915 and Skolem 1920): any first order theory in a countable language that has a model has a countable model.
- that can be counted
Example: countable sins
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As for set theory the failure of categoricity was already taken note of by Skolem in 1923 because it follows from the Löwenheim-Skolem Theorem which Skolem arrived at that year see Skolem 1923 based on Löwenheim 1915 and Skolem 1920 any first order theory in a countable language that has a model has a countable model