Ideas of William D. Hart, by Theme
[American, fl. 1994, At the University of Illinois, Chicago.]
green numbers give full details 
back to list of philosophers 
expand these ideas
1. Philosophy / C. History of Philosophy / 4. Later European Philosophy / c. Eighteenth century philosophy
13466

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

1. Philosophy / D. Nature of Philosophy / 5. Aims of Philosophy / d. Philosophy as puzzles
13477

The problems are the monuments of philosophy

1. Philosophy / F. Analytic Philosophy / 6. Logical Analysis
13515

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

3. Truth / C. Correspondence Truth / 2. Correspondence to Facts
13469

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

3. Truth / F. Semantic Truth / 1. Tarski's Truth / b. Satisfaction and truth
13504

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

3. Truth / F. Semantic Truth / 2. Semantic Truth
13503

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

4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / c. Derivation rules of PL
13500

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

4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / e. Existential quantifier ∃
13502

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

4. Formal Logic / F. Set Theory ST / 1. Set Theory
13456

Set theory articulates the concept of order (through relations)

13497

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

4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / a. Symbols of ST
13443

∈ relates across layers, while ⊆ relates within layers

4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
13442

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

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
13493

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

13495

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

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
13461

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

13462

With the Axiom of Choice every set can be wellordered

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / o. Axiom of Constructibility V = L
13516

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

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Naïve logical sets
13441

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

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
13494

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

4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
13457

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

13460

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

13458

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

13490

Von Neumann defines α<β as α∈β

4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
13481

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

5. Theory of Logic / G. Quantification / 3. Objectual Quantification
13506

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

5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
13512

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

13513

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

13505

Model theory studies how set theory can model sets of sentences

13511

Model theory is mostly confined to firstorder theories

5. Theory of Logic / K. Features of Logics / 6. Compactness
13496

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

5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / c. Berry's paradox
13484

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

5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / c. BuraliForti's paradox
13482

The BuraliForti paradox is a crisis for Cantor's ordinals

5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / a. The Liar paradox
13507

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

5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / b. The Heap paradox ('Sorites')
9117

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

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
13492

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

13459

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

13463

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

13491

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

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
13446

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

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
13509

We can establish truths about infinite numbers by means of induction

6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
13474

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

6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
13471

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

7. Existence / C. Structure of Existence / 8. Stuff / a. Pure stuff
13488

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

12. Knowledge Sources / A. A Priori Knowledge / 2. SelfEvidence
13480

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

12. Knowledge Sources / A. A Priori Knowledge / 11. Denying the A Priori
13476

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

18. Thought / D. Concepts / 3. Ontology of Concepts / c. Fregean concepts
13475

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