8667
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The 'integers' are the positive and negative natural numbers, plus zero [Friend]
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Full Idea:
The set of 'integers' is all of the negative natural numbers, and zero, together with the positive natural numbers.
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From:
Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5)
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A reaction:
Zero always looks like a misfit at this party. Credit and debit explain positive and negative nicely, but what is the difference between having no money, and money being irrelevant? I can be 'broke', but can the North Pole be broke?
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8671
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The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend]
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Full Idea:
The set of 'real' numbers, which consists of the rational numbers and the irrational numbers together, represents "the continuum", since it is like a smooth line which has no gaps (unlike the rational numbers, which have the irrationals missing).
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From:
Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5)
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A reaction:
The Continuum is the perfect abstract object, because a series of abstractions has arrived at a vast limit in its nature. It still has dizzying infinities contained within it, and at either end of the line. It makes you feel humble.
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15896
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Cantor needed Power Set for the reals, but then couldn't count the new collections [Cantor, by Lavine]
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Full Idea:
Cantor grafted the Power Set axiom onto his theory when he needed it to incorporate the real numbers, ...but his theory was supposed to be theory of collections that can be counted, but he didn't know how to count the new collections.
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From:
report of George Cantor (The Theory of Transfinite Numbers [1897]) by Shaughan Lavine - Understanding the Infinite I
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A reaction:
I take this to refer to the countability of the sets, rather than the members of the sets. Lavine notes that counting was Cantor's key principle, but he now had to abandon it. Zermelo came to the rescue.
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8663
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Raising omega to successive powers of omega reveal an infinity of infinities [Friend]
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Full Idea:
After the multiples of omega, we can successively raise omega to powers of omega, and after that is done an infinite number of times we arrive at a new limit ordinal, which is called 'epsilon'. We have an infinite number of infinite ordinals.
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From:
Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4)
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A reaction:
When most people are dumbstruck by the idea of a single infinity, Cantor unleashes an infinity of infinities, which must be the highest into the stratosphere of abstract thought that any human being has ever gone.
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8662
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The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend]
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Full Idea:
The first 'limit ordinal' is called 'omega', which is ordinal because it is greater than other numbers, but it has no immediate predecessor. But it has successors, and after all of those we come to twice-omega, which is the next limit ordinal.
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From:
Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.4)
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A reaction:
This is the gateway to Cantor's paradise of infinities, which Hilbert loved and defended. Who could resist the pleasure of being totally boggled (like Aristotle) by a concept such as infinity, only to have someone draw a map of it? See 8663 for sequel.
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