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6. Mathematics / A. Nature of Mathematics / 4. The Infinite / a. The Infinite

[the status and nature of infinity as a number]

26 ideas
Without infinity time has limits, magnitudes are indivisible, and numbers come to an end [Aristotle]
Postulate 2 says a line can be extended continuously [Shapiro on Euclid]
Not all infinites are equal [Newton]
A truly infinite quantity does not need to be a variable [Bolzano]
Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Lavine on Cantor]
It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Lavine on Cantor]
No one shall drive us out of the paradise the Cantor has created for us [Hilbert]
We extend finite statements with ideal ones, in order to preserve our logic [Hilbert]
Only the finite can bring certainty to the infinite [Hilbert]
Infinity and continuity used to be philosophy, but are now mathematics [Russell]
Zeno achieved the statement of the problems of infinitesimals, infinity and continuity [Russell]
We do not currently know whether, of two infinite numbers, one must be greater than the other [Russell]
There are cardinal and ordinal theories of infinity (while continuity is entirely ordinal) [Russell]
Infinity is not a number, so doesn't say how many; it is the property of a law [Wittgenstein]
Gödel showed that the syntactic approach to the infinite is of limited value [Kreisel]
Infinite natural numbers is as obvious as infinite sentences in English [Boolos]
Mathematics shows that thinking is not confined to the finite [Badiou]
We can establish truths about infinite numbers by means of induction [Hart,WD]
Cantor extended the finite (rather than 'taming the infinite') [Mayberry]
Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry]
Infinite numbers are qualitatively different - they are not just very large numbers [Heil]
Cantor's account of infinities has the shaky foundation of irrational numbers [Clegg]
The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine]
The infinite is extrapolation from the experience of indefinitely large size [Lavine]
Combining the concepts of negation and finiteness gives the concept of infinity [Jenkins]
Intuitionists only accept a few safe infinities [Colyvan]