denumerably

A positive by-product of Richard's paradox in Brouwer's work (1907, p. 149) is the notion of denumerably unfinished set, i.e., a set in which we can determine only denumerable subsets of elements, but where these denumerable subsets do not exhaust the given set, so that one can immediately produce new elements of the set from any given denumerable such subset.

Paradoxes and Contemporary Logic Cantini, Andrea 2007

Since an ordinary first-order language has a denumerable supply of open formulas, at most denumerably many sets (in any given domain) can be specified in this way.

Wild Dreams Of Reality, 3 2009

Obviously, a combinator that has a principal type, has denumerably many principal types, which are all substitution instances of each other; hence it is justified to talk about the principal type schema of a combinator.

Combinatory Logic Bimbó, Katalin 2008

Nonetheless, the intermediate Wittgenstein clearly rejects the notion that a non-denumerably infinite set is greater in cardinality than a denumerably infinite set.

Wittgenstein's Philosophy of Mathematics Rodych, Victor 2007

Typical examples of denumerably unfinished sets are the totality of countable ordinals, the points of the continuum, and in particular, as could be shown using Richard's paradox, the set of all definable points of the continuum (1907, p. 150).

Paradoxes and Contemporary Logic Cantini, Andrea 2007

According to Weyl, Richard's paradox teaches us the following distinction: on the one hand, we are able to characterize only denumerably many subsets of a given set by means of explicit definitions; but, on the other hand, new objects and (possibly uncountable) sets can be introduced by applying the remaining set theoretic operations, like power set or union.

Paradoxes and Contemporary Logic Cantini, Andrea 2007

(Tarski's remark that the supposition that all objects have names in the language can never be realized can be justified, for example, by observing that there are non-denumerably many sets of natural numbers, but in the languages he considers there are only denumerably many constants.)

Alfred Tarski Gómez-Torrente, Mario 2006

When an attempt (in a draft of the dissertation) at making constructive sense out of Cantor's second number class (the class of all denumerably infinite ordinals) and higher classes of even greater ordinals fails, he realises that this cannot be done and rejects the higher number classes, leaving only all finite ordinals and an unfinished or open-ended collection of denumerably infinite ordinals.

Luitzen Egbertus Jan Brouwer van Atten, Mark 2005

So understood there are denumerably many distinct quotation marks, each formed by prefixing a natural number n of strokes to a small circle, and in order to meaningfully enclose an expression with Boolos quotation marks we have to choose quotation marks of a ˜higher order™ than the quotation marks that occur in the expression to be quoted (if there are any) and each quotation mark in a grammatical sentence is to be ˜paired™ with the next identical quotation mark (Boolos 1995, p. 291).

Quotation Cappelen, Herman 2009

Proof: Although this theorem holds in general, we assume here that the set K of non-logical terminology is either finite or denumerably infinite (i.e., the size of the natural numbers, usually called

Classical Logic Shapiro, Stewart 2000