Ideas of Shaughan Lavine, by Theme

[American, fl. 2006, Professor at the University of Arizona.]

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4. Formal Logic / F. Set Theory ST / 1. Set Theory
Second-order set theory just adds a version of Replacement that quantifies over functions
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one
4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set
Collections of things can't be too big, but collections by a rule seem unlimited in size
4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
Those who reject infinite collections also want to reject the Axiom of Choice
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
The Power Set is just the collection of functions from one collection to another
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
Replacement was immediately accepted, despite having very few implications
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
Pure collections of things obey Choice, but collections defined by a rule may not
The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / c. Logical sets
The 'logical' notion of class has some kind of definition or rule to characterise the class
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
The iterative conception of set wasn't suggested until 1947
The iterative conception needs the Axiom of Infinity, to show how far we can iterate
The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement
Limitation of Size is not self-evident, and seems too strong
4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
A collection is 'well-ordered' if there is a least element, and all of its successors can be identified
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Second-order logic presupposes a set of relations already fixed by the first-order domain
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
Mathematical proof by contradiction needs the law of excluded middle
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity
6. Mathematics / A. Nature of Mathematics / 3. Numbers / b. Types of number
Every rational number, unlike every natural number, is divisible by some other number
6. Mathematics / A. Nature of Mathematics / 3. Numbers / g. Real numbers
For the real numbers to form a set, we need the Continuum Hypothesis to be true
6. Mathematics / A. Nature of Mathematics / 3. Numbers / h. Reals from Cauchy
Cauchy gave a necessary condition for the convergence of a sequence
Irrational numbers are the limits of Cauchy sequences of rational numbers
6. Mathematics / A. Nature of Mathematics / 3. Numbers / i. Reals from cuts
The two sides of the Cut are, roughly, the bounding commensurable ratios
6. Mathematics / A. Nature of Mathematics / 3. Numbers / p. Counting
Counting results in well-ordering, and well-ordering makes counting possible
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / a. The Infinite
The theory of infinity must rest on our inability to distinguish between very large sizes
The infinite is extrapolation from the experience of indefinitely large size
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / c. Potential infinite
The intuitionist endorses only the potential infinite
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / f. Uncountable infinities
'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / h. Ordinal infinity
Ordinals are basic to Cantor's transfinite, to count the sets
Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal
Cantor's theory concerns collections which can be counted, using the ordinals
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / i. Cardinal infinity
Paradox: there is no largest cardinal, but the class of everything seems to be the largest
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / a. Mathematics is set theory
Set theory will found all of mathematics - except for the notion of proof
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
Modern mathematics works up to isomorphism, and doesn't care what things 'really are'
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
Intuitionism rejects set-theory to found mathematics