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Single Idea 14133
[filed under theme 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
]
Full Idea
The theory of infinity has two forms, cardinal and ordinal, of which the former springs from the logical theory of numbers; the theory of continuity is purely ordinal.
Gist of Idea
There are cardinal and ordinal theories of infinity (while continuity is entirely ordinal)
Source
Bertrand Russell (The Principles of Mathematics [1903], §249)
Book Ref
Russell,Bertrand: 'Principles of Mathematics' [Routledge 1992], p.259
The
26 ideas
with the same theme
[the status and nature of infinity as a number]:
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18090
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Without infinity time has limits, magnitudes are indivisible, and numbers come to an end
[Aristotle]
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7555
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Zeno achieved the statement of the problems of infinitesimals, infinity and continuity
[Russell on Zeno of Citium]
|
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8738
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Postulate 2 says a line can be extended continuously
[Euclid, by Shapiro]
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13151
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Not all infinites are equal
[Newton]
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10856
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A truly infinite quantity does not need to be a variable
[Bolzano]
|
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15902
|
Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties
[Cantor, by Lavine]
|
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15908
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It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers
[Cantor, by Lavine]
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9633
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No one shall drive us out of the paradise the Cantor has created for us
[Hilbert]
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12460
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We extend finite statements with ideal ones, in order to preserve our logic
[Hilbert]
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12462
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Only the finite can bring certainty to the infinite
[Hilbert]
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14420
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Infinity and continuity used to be philosophy, but are now mathematics
[Russell]
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14119
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We do not currently know whether, of two infinite numbers, one must be greater than the other
[Russell]
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14133
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There are cardinal and ordinal theories of infinity (while continuity is entirely ordinal)
[Russell]
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18708
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Infinity is not a number, so doesn't say how many; it is the property of a law
[Wittgenstein]
|
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17809
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Gödel showed that the syntactic approach to the infinite is of limited value
[Kreisel]
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10491
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Infinite natural numbers is as obvious as infinite sentences in English
[Boolos]
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9813
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Mathematics shows that thinking is not confined to the finite
[Badiou]
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13509
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We can establish truths about infinite numbers by means of induction
[Hart,WD]
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17799
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Cantor's infinite is an absolute, of all the sets or all the ordinal numbers
[Mayberry]
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17797
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Cantor extended the finite (rather than 'taming the infinite')
[Mayberry]
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18518
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Infinite numbers are qualitatively different - they are not just very large numbers
[Heil]
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10866
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Cantor's account of infinities has the shaky foundation of irrational numbers
[Clegg]
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15949
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The theory of infinity must rest on our inability to distinguish between very large sizes
[Lavine]
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15947
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The infinite is extrapolation from the experience of indefinitely large size
[Lavine]
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17730
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Combining the concepts of negation and finiteness gives the concept of infinity
[Jenkins]
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17923
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Intuitionists only accept a few safe infinities
[Colyvan]
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