8 Sets HL

8 Sets
Note:8@@@@@@

Yellow highlight | Location: 2,272

8.1   Sets are not collections
Note:Tttttttttt

Yellow highlight | Location: 2,273

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

Yellow highlight | Location: 2,276

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

Yellow highlight | Location: 2,280

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

Yellow highlight | Location: 2,282

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

Yellow highlight | Location: 2,286

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

Yellow highlight | Location: 2,290

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

Yellow highlight | Location: 2,297

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

Yellow highlight | Location: 2,300

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

Yellow highlight | Location: 2,307

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

Yellow highlight | Location: 2,310

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

Yellow highlight | Location: 2,311

8.2   Sets are not defined by the axioms
Note:Tttttttttttt

Yellow highlight | Location: 2,313

sets are simply the things that satisfy those axioms.
Note:ALTRA SPIEGA POPOLARE

Yellow highlight | Location: 2,315

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

Yellow highlight | Location: 2,316

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

Yellow highlight | Location: 2,348

8.3   Many regarded as one: the foundational sin?
Note:Tttttttttt

Yellow highlight | Location: 2,349

‘A set is a many which allows itself to be thought of as a one.’
Note:LA DEF DI CANTOE

Yellow highlight | Location: 2,351

this rules out the empty set,
Note:PRIMO

Yellow highlight | Location: 2,352

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

Yellow highlight | Location: 2,354

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...

Yellow highlight | Location: 2,356

Two does not equal one,
Note:Cccccc

Yellow highlight | Location: 2,358

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

Yellow highlight | Location: 2,375

8.4   The significance of the paradoxes
Note:Tttttttttt

Yellow highlight | Location: 2,376

Naive Comprehension Axiom
Note:COS È

Yellow highlight | Location: 2,377

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

Yellow highlight | Location: 2,379

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

Yellow highlight | Location: 2,379

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

Yellow highlight | Location: 2,383

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

Yellow highlight | Location: 2,386

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

Yellow highlight | Location: 2,396

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

Yellow highlight | Location: 2,404

8.5   Are numbers sets?
Note:Tttttttt

Yellow highlight | Location: 2,406

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

Yellow highlight | Location: 2,408

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

Note | Location: 2,410

LA FONDAZIONE VINSIEMISTICA DELL ARITMETICA DI VRUSSELL

Yellow highlight | Location: 2,416

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

Yellow highlight | Location: 2,439

the constructions are wildly implausible on their face
Note:CONCLUSIONE

Yellow highlight | Location: 2,442

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

Yellow highlight | Location: 2,444

8.6   Set theory and the laws of arithmetic
Note:Ttttttttt

Yellow highlight | Location: 2,446

+ b = b + a    (Commutative Law of
Note:ESEMPIO DI LEGGI ARITMETICHE

Yellow highlight | Location: 2,453

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

Yellow highlight | Location: 2,454

This answer is crazily implausible.
Note:GIUDIZIO

Yellow highlight | Location: 2,454

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

Yellow highlight | Location: 2,457

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

Yellow highlight | Location: 2,462

‘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

Yellow highlight | Location: 2,466

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

Yellow highlight | Location: 2,472

‘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

Yellow highlight | Location: 2,474

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

Yellow highlight | Location: 2,476

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...

Yellow highlight | Location: 2,477

Conclusion: set theory is not the foundation of arithmetic,
CONCLUSIONE @@@@@@@@