Bijection

You can make each point have measure 1/2, for example, or assign whatever positive weights you like to each point (such measures are in bijection with functions from your base set to the positive reals). ❋ Unknown (2007)

If we are willing make the further assumption that it only takes one bijection to one such instance of the power set of ω to render the power set itself “absolutely” countable, then we can understand the Skolemite's strong claim about absolute countability. ❋ Bays, Timothy (2009)

Ω (mˆ) says that there is no bijection between the natural numbers and mˆ. ❋ Bays, Timothy (2009)

(A set is reflexive iff it is equipollent to one of its proper subsets; and two sets are equipollent with one another iff there exists a bijection, i.e., a one-to-one correspondence, between them.) ❋ Gerard Varni (2009)

If asked about the phrase “is a bijection,” she will go on to talk about collections of ordered pairs satisfying certain nice properties, and if asked about the term ❋ Bays, Timothy (2009)

Dedekind also provided a proof of the Cantor-Bernstein Theorem (that between any two sets which can be embedded into each other one-to-one there exists a bijection, so that they have the same cardinality), another basic result in the modern theory of transfinite cardinals. ❋ Reck, Erich (2008)

The bijection we have just observed can now be stated as ❋ Pratt, Vaughan (2007)

F is a bijection from W onto the set of all maximal consistent sets of facts. ❋ Mulligan, Kevin (2007)

F is a bijection from W onto the set of all conjunctively complete sets of facts; ❋ Mulligan, Kevin (2007)

φ is a bijection from the set of all singletons* onto the set of all conjunctively complete sets of facts; ❋ Mulligan, Kevin (2007)

φ is a bijection from the set of all singletons* onto the set of all maximal consistent sets of facts. ❋ Mulligan, Kevin (2007)

F is a bijection from W onto the set of all sets of facts; ❋ Mulligan, Kevin (2007)

φ is a bijection from the set of all singletons* onto the set of all sets of facts; ❋ Mulligan, Kevin (2007)

On the other hand, in Mirimanoff's 1917a there is a remarkable use of Burali-Forti's paradox which suggests a necessary condition for set-hood in terms of size, viz., if a collection is in bijection with the set of all ordinals, then it does not exist as a set. ❋ Cantini, Andrea (2007)

It goes as follows: start with a model of something like ZF, and a bijection k between V and V × {0, 1}. ❋ Forster, Thomas (2006)

But when we were talking about continued fractions, all we cared about was sequences, and we can define a sequence to be infinite if its terms can be put in bijection with the natural numbers. ❋ Gowers (2010)

And because Cantor's proof is constructive, we can even go one step further and see how it refutes their alleged bijection. ❋ Unknown (2010)

In the case of Cantor's diagonal argument, giving a bijection between N and R would of course refute Cantor, end of discussion. ❋ Unknown (2010)

Now I have to show there is a bijection between S and the natural numbers Z+.. ❋ Unknown (2009)

The way to understand this bijection is to label the edges with "rules for changing face colorings." ❋ Unknown (2009)