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15924
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Predicative definitions are acceptable in mathematics if they distinguish objects, rather than creating them? [Zermelo, by Lavine]
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Full Idea:
On Zermelo's view, predicative definitions are not only indispensable to mathematics, but they are unobjectionable since they do not create the objects they define, but merely distinguish them from other objects.
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From:
report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Shaughan Lavine - Understanding the Infinite V.1
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A reaction:
This seems to have an underlying platonism, that there are hitherto undefined 'objects' lying around awaiting the honour of being defined. Hm.
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19463
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Induction assumes some uniformity in nature, or that in some respects the future is like the past [Ayer]
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Full Idea:
In all inductive reasoning we make the assumption that there is a measure of uniformity in nature; or, roughly speaking, that the future will, in the appropriate respects, resemble the past.
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From:
A.J. Ayer (The Problem of Knowledge [1956], 2.viii)
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A reaction:
I would say that nature is 'stable'. Nature changes, so a global assumption of total uniformity is daft. Do we need some global uniformity assumptions, if the induction involved is local? I would say yes. Are all inductions conditional on this?
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17608
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We take set theory as given, and retain everything valuable, while avoiding contradictions [Zermelo]
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Full Idea:
Starting from set theory as it is historically given ...we must, on the one hand, restrict these principles sufficiently to exclude as contradiction and, on the other, take them sufficiently wide to retain all that is valuable in this theory.
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From:
Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908], Intro)
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A reaction:
Maddy calls this the one-step-back-from-disaster rule of thumb. Zermelo explicitly mentions the 'Russell antinomy' that blocked Frege's approach to sets.
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17607
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Set theory investigates number, order and function, showing logical foundations for mathematics [Zermelo]
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Full Idea:
Set theory is that branch whose task is to investigate mathematically the fundamental notions 'number', 'order', and 'function', taking them in their pristine, simple form, and to develop thereby the logical foundations of all of arithmetic and analysis.
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From:
Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908], Intro)
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A reaction:
At this point Zermelo seems to be a logicist. Right from the start set theory was meant to be foundational to mathematics, and not just a study of the logic of collections.
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13015
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Zermelo used Foundation to block paradox, but then decided that only Separation was needed [Zermelo, by Maddy]
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Full Idea:
Zermelo used a weak form of the Axiom of Foundation to block Russell's paradox in 1906, but in 1908 felt that the form of his Separation Axiom was enough by itself, and left the earlier axiom off his published list.
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From:
report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.2
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A reaction:
Foundation turns out to be fairly controversial. Barwise actually proposes Anti-Foundation as an axiom. Foundation seems to be the rock upon which the iterative view of sets is built. Foundation blocks infinite descending chains of sets, and circularity.
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13027
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Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets [Zermelo, by Maddy]
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Full Idea:
Zermelo was a reductionist, and believed that theorems purportedly about numbers (cardinal or ordinal) are really about sets, and since Von Neumann's definitions of ordinals and cardinals as sets, this has become common doctrine.
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From:
report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by Penelope Maddy - Believing the Axioms I §1.8
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A reaction:
Frege has a more sophisticated take on this approach. It may just be an updating of the Greek idea that arithmetic is about treating many things as a unit. A set bestows an identity on a group, and that is all that is needed.
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9627
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Different versions of set theory result in different underlying structures for numbers [Zermelo, by Brown,JR]
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Full Idea:
In Zermelo's set-theoretic definition of number, 2 is a member of 3, but not a member of 4; in Von Neumann's definition every number is a member of every larger number. This means they have two different structures.
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From:
report of Ernst Zermelo (Investigations in the Foundations of Set Theory I [1908]) by James Robert Brown - Philosophy of Mathematics Ch. 4
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A reaction:
This refers back to the dilemma highlighted by Benacerraf, which was supposed to be the motivation for structuralism. My intuition says that the best answer is that they are both wrong. In a pattern, the nodes aren't 'members' of one another.
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19459
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To say 'I am not thinking' must be false, but it might have been true, so it isn't self-contradictory [Ayer]
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Full Idea:
To say 'I am not thinking' is self-stultifying since if it is said intelligently it must be false: but it is not self-contradictory. The proof that it is not self-contradictory is that it might have been false.
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From:
A.J. Ayer (The Problem of Knowledge [1956], 2.iii)
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A reaction:
If it doesn't imply a contradiction, then it is not a necessary truth, which is what it is normally taken to be. Is 'This is a sentence' necessarily true? It might not have been one, if the rules of English syntax changed recently.
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19460
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'I know I exist' has no counterevidence, so it may be meaningless [Ayer]
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Full Idea:
If there is no experience at all of finding out that one is not conscious, or that one does not exist, ..it is tempting to say that sentences like 'I exist', 'I am conscious', 'I know that I exist' do not express genuine propositions.
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From:
A.J. Ayer (The Problem of Knowledge [1956], 2.iii)
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A reaction:
This is, of course, an application of the somewhat discredited verification principle, but the fact that strictly speaking the principle has been sort of refuted does not mean that we should not take it seriously, and be influenced by it.
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19462
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Induction passes from particular facts to other particulars, or to general laws, non-deductively [Ayer]
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Full Idea:
Inductive reasoning covers all cases in which we pass from a particular statement of fact, or set of them, to a factual conclusion which they do not formally entail. The inference may be to a general law, or by analogy to another particular instance.
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From:
A.J. Ayer (The Problem of Knowledge [1956], 2.viii)
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A reaction:
My preferred definition is 'learning from experience' - which I take to be the most rational behaviour you could possibly imagine. I don't think a definition should be couched in terms of 'objects' or 'particulars'.
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