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Ideas of William D. Hart, by Text
[American, fl. 1994, At the University of Illinois, Chicago.]
1992

HatTricks and Heaps


p.2

9117

The smallest heap has four objects: three on the bottom, one on the top [Sorensen]

2010

The Evolution of Logic


p.22

13456

Set theory articulates the concept of order (through relations)

1

p.4

13442

Without the empty set we could not form a∩b without checking that a and b meet

1

p.4

13441

Naïve set theory has trouble with comprehension, the claim that every predicate has an extension

1

p.5

13443

∈ relates across layers, while ⊆ relates within layers

1

p.18

13446

19th century arithmetization of analysis isolated the real numbers from geometry

1

p.23

13457

A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets

1

p.23

13460

'Wellordering' must have a least member, so it does the natural numbers but not the integers

1

p.23

13458

A partial ordering becomes 'total' if any two members of its field are comparable

1

p.26

13459

The lessthan relation < wellorders, and partially orders, and totally orders the ordinal numbers

1

p.27

13463

There are at least as many infinite cardinals as transfinite ordinals (because they will map)

1

p.27

13461

We can choose from finite and evident sets, but not from infinite opaque ones

1

p.27

13462

With the Axiom of Choice every set can be wellordered

10

p.268

13515

To study abstract problems, some knowledge of set theory is essential

10

p.270

13516

If we accept that V=L, it seems to settle all the open questions of set theory

2

p.31

13466

We are all postKantians, because he set the current agenda for philosophy

2

p.36

13469

Tarski showed how we could have a correspondence theory of truth, without using 'facts'

2

p.41

13471

Mathematics makes existence claims, but philosophers usually say those are never analytic

2

p.44

13474

Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several

2

p.47

13475

The Fregean concept of GREEN is a function assigning true to green things, and false to the rest

2

p.53

13476

The failure of key assumptions in geometry, mereology and set theory throw doubt on the a priori

2

p.53

13477

The problems are the monuments of philosophy

2

p.58

13481

Maybe sets should be rethought in terms of the even more basic categories

3

p.59

13482

The BuraliForti paradox is a crisis for Cantor's ordinals

3

p.63

13484

Berry's Paradox: we succeed in referring to a number, with a term which says we can't do that

3

p.71

13488

Mass words do not have plurals, or numerical adjectives, or use 'fewer'

3

p.74

13490

Von Neumann defines α<β as α∈β

3

p.74

13491

The axiom of infinity with separation gives a least limit ordinal ω

3

p.75

13492

Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton

3

p.79

13493

In the modern view, foundation is the heart of the way to do set theory

3

p.80

13495

Foundation Axiom: an nonempty set has a member disjoint from it

3

p.80

13494

The iterative conception may not be necessary, and may have fixed points or infinitely descending chains

3

p.80

13496

Firstorder logic is 'compact': consequences of a set are consequences of a finite subset

3

p.88

13497

Nowadays ZFC and NBG are the set theories; types are dead, and NF is only useful for the whole universe

4

p.90

13500

Conditional Proof: infer a conditional, if the consequent can be deduced from the antecedent

4

p.96

13502

∃y... is read as 'There exists an individual, call it y, such that...', and not 'There exists a y such that...'

4

p.101

13503

A firstorder language has an infinity of Tsentences, which cannot add up to a definition of truth

4

p.107

13504

Truth for sentences is satisfaction of formulae; for sentences, either all sequences satisfy it (true) or none do

4

p.108

13505

Model theory studies how set theory can model sets of sentences

4

p.111

13506

The universal quantifier can't really mean 'all', because there is no universal set

4

p.122

13507

The machinery used to solve the Liar can be rejigged to produce a new Liar

5

p.144

13509

We can establish truths about infinite numbers by means of induction

9

p.236

13511

Model theory is mostly confined to firstorder theories

9

p.238

13512

Modern model theory begins with the proof of Los's Conjecture in 1962

9

p.238

13513

Models are ways the world might be from a firstorder point of view

p.350

p.350

13480

Fregean selfevidence is an intrinsic property of basic truths, rules and definitions
