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Single Idea 13509
[filed under theme 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
]
Full Idea
Mathematical induction is a way to establish truths about the infinity of natural numbers by a finite proof.
Gist of Idea
We can establish truths about infinite numbers by means of induction
Source
William D. Hart (The Evolution of Logic [2010], 5)
Book Ref
Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.144
A Reaction
If there are truths about infinities, it is very tempting to infer that the infinities must therefore 'exist'. A nice, and large, question in philosophy is whether there can be truths without corresponding implications of existence.
The
26 ideas
with the same theme
[the status and nature of infinity as a number]:
18090
|
Without infinity time has limits, magnitudes are indivisible, and numbers come to an end
[Aristotle]
|
7555
|
Zeno achieved the statement of the problems of infinitesimals, infinity and continuity
[Russell on Zeno of Citium]
|
8738
|
Postulate 2 says a line can be extended continuously
[Euclid, by Shapiro]
|
13151
|
Not all infinites are equal
[Newton]
|
10856
|
A truly infinite quantity does not need to be a variable
[Bolzano]
|
15902
|
Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties
[Cantor, by Lavine]
|
15908
|
It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers
[Cantor, by Lavine]
|
9633
|
No one shall drive us out of the paradise the Cantor has created for us
[Hilbert]
|
12460
|
We extend finite statements with ideal ones, in order to preserve our logic
[Hilbert]
|
12462
|
Only the finite can bring certainty to the infinite
[Hilbert]
|
14420
|
Infinity and continuity used to be philosophy, but are now mathematics
[Russell]
|
14119
|
We do not currently know whether, of two infinite numbers, one must be greater than the other
[Russell]
|
14133
|
There are cardinal and ordinal theories of infinity (while continuity is entirely ordinal)
[Russell]
|
18708
|
Infinity is not a number, so doesn't say how many; it is the property of a law
[Wittgenstein]
|
17809
|
Gödel showed that the syntactic approach to the infinite is of limited value
[Kreisel]
|
10491
|
Infinite natural numbers is as obvious as infinite sentences in English
[Boolos]
|
9813
|
Mathematics shows that thinking is not confined to the finite
[Badiou]
|
13509
|
We can establish truths about infinite numbers by means of induction
[Hart,WD]
|
17797
|
Cantor extended the finite (rather than 'taming the infinite')
[Mayberry]
|
17799
|
Cantor's infinite is an absolute, of all the sets or all the ordinal numbers
[Mayberry]
|
18518
|
Infinite numbers are qualitatively different - they are not just very large numbers
[Heil]
|
10866
|
Cantor's account of infinities has the shaky foundation of irrational numbers
[Clegg]
|
15947
|
The infinite is extrapolation from the experience of indefinitely large size
[Lavine]
|
15949
|
The theory of infinity must rest on our inability to distinguish between very large sizes
[Lavine]
|
17730
|
Combining the concepts of negation and finiteness gives the concept of infinity
[Jenkins]
|
17923
|
Intuitionists only accept a few safe infinities
[Colyvan]
|