Combining Philosophers
Ideas for Moses Schönfinkel, Richard Dedekind and Kurt Gdel
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18 ideas
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
10132
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There can be no single consistent theory from which all mathematical truths can be derived [Gödel, by George/Velleman]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
9823
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Numbers are free creations of the human mind, to understand differences [Dedekind]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
10090
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Dedekind defined the integers, rationals and reals in terms of just the natural numbers [Dedekind, by George/Velleman]
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7524
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Order, not quantity, is central to defining numbers [Dedekind, by Monk]
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17452
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Ordinals can define cardinals, as the smallest ordinal that maps the set [Dedekind, by Heck]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
14131
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Dedekind's ordinals are just members of any progression whatever [Dedekind, by Russell]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
17611
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We want the essence of continuity, by showing its origin in arithmetic [Dedekind]
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
14437
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Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Dedekind, by Russell]
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18094
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Dedekind says each cut matches a real; logicists say the cuts are the reals [Dedekind, by Bostock]
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18244
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I say the irrational is not the cut itself, but a new creation which corresponds to the cut [Dedekind]
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10572
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A cut between rational numbers creates and defines an irrational number [Dedekind]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
9824
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In counting we see the human ability to relate, correspond and represent [Dedekind]
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6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic
17612
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Arithmetic is just the consequence of counting, which is the successor operation [Dedekind]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / b. Mark of the infinite
9826
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A system S is said to be infinite when it is similar to a proper part of itself [Dedekind]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
10868
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The Continuum Hypothesis is not inconsistent with the axioms of set theory [Gödel, by Clegg]
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13517
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If set theory is consistent, we cannot refute or prove the Continuum Hypothesis [Gödel, by Hart,WD]
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10046
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The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers [Gödel]
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6. Mathematics / A. Nature of Mathematics / 5. The Infinite / l. Limits
18087
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If x changes by less and less, it must approach a limit [Dedekind]
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