venerdì 24 agosto 2018

8 Sets HL

8 Sets
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8.1   Sets are not collections
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According to the now-standard view developed by the likes of Cantor, Frege, and Russell, all of mathematics is based on set theory.
Note:STANDARD VIEW

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The most common explanation given is that a set is simply a collection or group.
Note:DEF....POCO CHIARA PER H

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It is uncontroversial in set theory that there is an ‘empty set’, a set with no members. What collection is this supposed to be?
Note:ESEMPIO CHE CONTRADDICE

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It is uncontroversial in set theory that there are singleton sets,
Note:ALTRO ES CONTRADDITTORIO

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It is again uncontroversial in set theory that there are infinitely many ‘pure sets’, that is, sets that are constructed, from the ground up, using no objects other than sets.
Note:ALTRO ES PROBLEMATICO

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there are infinitely many infinite collections built up from nothing but other collections,
Note:ANALOGIA NONSENSE

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In set theory, sets are typically understood as abstract, non-physical objects, even when their members are physical.
Note:ALTRO ELEMENTO PROBLEMATICO

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a deer might be killed by a pack of wolves, but no deer is killed by an abstract object;
Note:ANALOGIA CHE CONFUTA

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My reason for skepticism about set theory is not anything to do with ‘simplicity’ or ‘weirdness’,
Note:TUTTAVIA

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I am saying we should be skeptical because no one has been able to explain
Note:PIUTTOSTO

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8.2   Sets are not defined by the axioms
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sets are simply the things that satisfy those axioms.
Note:ALTRA SPIEGA POPOLARE

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I don’t see that there is in reality anything having the characteristics mentioned in points (i)–(v) above.
Note:ASTRAZIONI MA NN IMMANENTI

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Second, the claim that the axioms of set theory define the term ‘set’ is refuted by a famous result in model theory: the Löwenheim-Skolem Theorem.
Note:QUANTO ALLA SECONDA DEFINIZIONE

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8.3   Many regarded as one: the foundational sin?
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‘A set is a many which allows itself to be thought of as a one.’
Note:LA DEF DI CANTOE

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this rules out the empty set,
Note:PRIMO

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It also rules out singletons, since a single object isn’t a many either.
Note:ALTRA LACUNA

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assume we have two or more objects. Cantor’s suggestion appears to be that (at least in most cases), it is permissible to regard these objects as one.
Note:MA CONCEDIAMO PURE...

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Two does not equal one,
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When we form the set {a, b}, we do not literally treat the two objects as one – we do not find ourselves saying that a = b. Rather, we treat {a, b} as a third thing,
Note:PIUTTOSTO

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8.4   The significance of the paradoxes
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Naive Comprehension Axiom
Note:COS È

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for any well-formed predicate, there is a set of the things that satisfy it.
Note:ESEMPIO DI PREDICATO: È ROSSO

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this axiom generates both Russell’s Paradox and Cantor’s Paradox.
Note:PURTROPPO

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there is something wrong with the notion of a set.
Note:CONCLUSIONE

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If a concept generates paradoxes, that is generally a reason for thinking it an invalid concept.
Note:DICIAMOLO CHIARAMENTE

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If the foundational intuitions on the basis of which some objects were initially introduced are proven to be contradictory, this removes the central reason we had for believing in those objects.
Note:CRITICA RADICALE

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the term ‘set’, according to modern set theory, has no associated set – there is no set of all the things it applies to, since that would have to be the set of all sets. Therefore, it seems, the term ‘set’ is meaningless.
Note:IL PARADOSSO DECISIVO

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8.5   Are numbers sets?
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If we reject sets, won’t we have to give up the rest of mathematics?
Note:VISTO CHE OGGI L INSIEMISTICA È IL FONDAMENTO DELLA MATEMATICA

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The number zero is just the empty set. The number one is the set of all sets that are equinumerous with {0}, where two sets are defined to be ‘equinumerous’ if and only if there exists a one-to-one function from either set onto the other. The number two is the set of all sets that are equinumerous, in that same sense, with {0,1}.
Note:LA FONDAZIONE RUSSELIANA DEI NUMERI...GRAZIE AGLI INSIEMI

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LA FONDAZIONE VINSIEMISTICA DELL ARITMETICA DI VRUSSELL

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let me explain fractions.
Note:MEGLIO DI NO!!!!!

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the constructions are wildly implausible on their face
Note:CONCLUSIONE

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If Cantor, Russell, et al., are merely saying, ‘Here are some objects that have the same formal properties as the number system (because I deliberately constructed them that way)’, I suppose this might be mildly interesting. But it hardly justifies claims to have identified the foundation of mathematics.
Note:COROLLARIO

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8.6   Set theory and the laws of arithmetic
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+ b = b + a    (Commutative Law of
Note:ESEMPIO DI LEGGI ARITMETICHE

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How do we know these things? Here is one answer: because they can be derived using one of the set-theoretic constructions for the numbers.
Note:ORIGINE ALTAMENTE IMPROB. DELLE LEGGI AROTMETICHE X RUSSELL

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This answer is crazily implausible.
Note:GIUDIZIO

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Derivations using set theoretic constructions tend to be complex and difficult to follow; it is much harder to see that they are correct than it is to see that the above laws are correct.
Note:DI FATTO

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When someone wants to argue that we ‘can construe’ the numbers 0, 1, 2, ... as the sets ∅, {∅}, {∅, {∅}}, ... , they do this by arguing that those sets have the formal properties of the natural numbers – that is, the properties that we already know the numbers have.
Note:PIÙ PROBABILE L OPPOSTO: L INSIEMISTICA QUADRA XCHÈ COSTRUITA SUI NUMERI

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‘Let us try, therefore, whether we can derive from our definition [ ... ] any of the well-known properties of numbers.’
Note:LE PAROLE CON CUI FREGE SI TRADISCE...LE WELL KNOWN PROPERTY

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it directly and obviously entails that what they are working on is not actually the foundations of mathematics, since the propositions that they seek to derive are known prior to and (epistemically) independently of the derivations.
Note:QUEL CHE SI VORREBBE DIMOSTRARE È GIÀ NOTO PRIMA COME CERTO...ED È QS CCERTEZZA CHE XAVVALORA LA DIMOSTRAZ

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‘Perhaps the set theoretic constructions explain, not how we come to know the truths of arithmetic, but rather what makes them true.
Note:UNO È TENTATO DI DIRE

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there is no plausible way in which we could know those facts independently of knowing anything about set theory.
Note:MA ANCHE QS È IMPLAUSIBILE

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there is no plausible way in which people could have known that without knowing any set theory and without doing any derivations using it.
Note:TRADOTTO...

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Conclusion: set theory is not the foundation of arithmetic,
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