Combining Texts
Ideas for
'On the Question of Absolute Undecidability', 'Of the First Principles of Government' and 'Infinity: Quest to Think the Unthinkable'
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12 ideas
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
10880
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Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) [Clegg]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
10860
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An ordinal number is defined by the set that comes before it [Clegg]
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10861
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Beyond infinity cardinals and ordinals can come apart [Clegg]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
10854
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Transcendental numbers can't be fitted to finite equations [Clegg]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / k. Imaginary numbers
10858
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By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line [Clegg]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / l. Zero
10853
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Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless [Clegg]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
10866
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Cantor's account of infinities has the shaky foundation of irrational numbers [Clegg]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
10869
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The Continuum Hypothesis is independent of the axioms of set theory [Clegg]
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10862
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The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
17890
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There are at least eleven types of large cardinal, of increasing logical strength [Koellner]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
17887
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PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
17891
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Arithmetical undecidability is always settled at the next stage up [Koellner]
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