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Logic and Set Theory

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**Actual infinity Aristotle distinguished actual vs potential infinities**

actual infinity: elements exist together simultaneously potential : elements exist only consecutively over time. In mathematics, actual infinity is the notion that all numbers (natural, real etc.) can be enumerated in some sense sufficiently definite for them to form a set. the abstraction of actual infinity involves the acceptance of infinite entities, such as the set of all natural numbers as given objects The term "actual" in is synonymous with definite, completed, extended or existential, but not to be mistaken for physically existing.

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**Gauss’ (1831) summary of common view:**

Gauss’ (1831) summary of common view: * I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction.. The drastic change was initialized by Bolzano and Cantor in the 19th century Bolzano (c. 1830s) introduced the notion of “set” * A multitude … with the property that every finite set … is only a part of it, I will call an infinite multitude Georg Cantor (c. 1870s) developed “set theory”; distinguished 3 realms of infinity: infinity of God, of nature, and of mathematics. * The numbers are a free creation of human mind. (R. Dedekind)

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Infinity In mathematical analysis, it is absolutely essential to deal with infinite quantities in a rigorous way. E.g., there are infinitely many numbers between 0 and 1, but we can say that the length of the interval [0,1] equals 1? Can we say that 0.999…=1? Can we say that ½+1/4+1/8+…=1? The natural numbers 1,2,3… are infinite The rational numbers are equal in number to the whole numbers. The real numbers exceed the number of integers

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**Countable Uncountable sets**

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**Real numbers and decimal and binary representations**

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Equivalence of sets Two sets A and B are equivalent if there is a rule that assigns to each element of A a unique element of B In a sense, equivalent sets have the same number of elements Continuum hypothesis: There are no sets that have more elements than the whole numbers and fewer than the real numbers.

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Logic and Set Theory How do we deduce mathematical theorems that depend on properties of sets? Boole: Boolean logic DeMorgan: Laws of inclusion and exclusion

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**Propositional calculus**

Assign truth values to “atomic” propositions Calculate truth of a compound statement based on truth values of atoms

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**Boole (1815-1864) and DeMorgan (1806–1871)**

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Boolean algebra

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**Boole (1815-1864) and DeMorgan (1806–1871)**

De Morgan’s laws: not (P and Q) = (not P) or (not Q) not (P or Q) = (not P) and (not Q)

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**Modus ponens ("mode that affirms”)**

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Truth table version

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**Other logical deduction rules**

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Continuum hypothesis There is no set whose size is strictly between that of the integers and that of the real numbers. CH and AC are consistent with ZF

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**Logic and infinite sets**

Problems arise when we try to apply predicate logic to statements about “all of the members of a set” “Second order logic”

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Russell’s paradox Suppose that, for any formal criterion, a set S exists whose members are exactly those objects satisfying the criterion Can’t do this IF a there is set S containing exactly the sets that are not members of themselves. If S qualifies as a member of itself, it would contradict its own definition as a set containing sets that are not members of themselves. If S is not a member of itself, it would qualify as a member of itself by the same definition. This contradiction is Russell's paradox.

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**The axiom of choice (AC)**

AC says that given any collection of bins, each containing at least one object, it is possible to select exactly one object from each bin, even if there are infinitely many bins and there is no "rule" for which object to pick from each.

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The Banach–Tarski paradox states that a solid ball in 3-dimensional space can be split into several non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball. Unlike most theorems in geometry, it depends in a critical way on the axiom of choice in set theory.

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Prisoners hat problem Prisoners are to be lined up facing forward. Each wears a black hat, or a white (random) Warden starts from the back of the line, asking each prisoner his hat color, then moving forward one by one. Prisoners are only allowed to say "black" or "white." They are executed if wrong and spared if right. Each prisoner can hear the guesses of other prisoners (*and their outcome). The prisoners learn of the warden's plan and the rules described above the night before this is to take place and can therefore strategize beforehand. What is their best plan? instead of a finite number of prisoners, take a countably infinite number of them (say, one for each natural number). Any attempt to extend the solution for the previous problem to this one results in an infinite number of executions.

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**Completeness Kurt Gödel in 1931, First incompleteness theorem:**

Any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory. Analogy: liar paradox: "This sentence is false."

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