Frege observes that this is the case if and only if (henceforth: iff) the concepts are equinumerous
in the sense that the objects falling under one concept can be one-one correlated to the objects falling under the other.
For a variety of sequences s, natural statistics on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] have the same distribution as natural statistics on other equinumerous
combinatorial families, a phenomenon that was used to settle an open question about Coxeter groups in .
Notice that this last proof shows that the three sets [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (321, 231), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](321, 312) and[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](321,231,312) are identical, not just equinumerous
. To my knowledge, there is no analogous result in the enumeration of general (as opposed to irreducible) permutations.
The training dataset was randomly divided into ten distinct equinumerous
subsets, for the purpose of the k-fold method.
Compare this: when it seems to us that a set could not possibly be equinumerous
with a proper subset, this seeming has no apparent correlated experience distinct from it.
To see that Cantor did not intend any restriction to definable sets, one only has to consider his remarks on the "internal determinateness" of the question whether two sets are equinumerous
. (70) From the discussion it is clear that the "law" by which elements of the two sets are put in correspondence need not be definable.
(16.) This leads to the familiar proofs that the set of even numbers is equinumerous
with the set of odd numbers, and that the natural numbers are equinumerous
with the rationals.
Part (2) of the BusyCode-Motion Lemma shows that [f.sub.CM,p]is onto; by the computational optimality of BCM (OCM Theorem 3.13), the domain and range of [f.sub.CM,p] are equinumerous
, so it must be one-to-one as well.
Yet again, two stroke string inscriptions are inscriptions of the same stroke string type if and only if the strokes which make up the inscriptions are equinumerous
. But if the imagined inscriptions consist of no definite number of strokes such that we could tell what that number is, it is uncertain what an equinumerous
relation here could come to.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is shown to be equinumerous
to such paths with [??] = 2 (Theorem 2.4), and more generally [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is equinumerous
to such paths with [??] = k (Theorem 2.3).
Clearly the members of either set not containing an occurrence of 113 or 133, respectively, are equinumerous
by the preceding.
He can, however, be credited with the following: he derived (or showed how to derive) the whole of arithmetic in second-order logic from "Hume's principle" ("the numbers belonging to F and G are equal if and only if F is equinumerous
with G"); he derived Peano's second postulate ("every natural number has a successor") from the same principle; he succeeded, where Dedekind failed, in demonstrating the existence of an infinite system (p.