PROBLEM SET 7
... have the same number of hairs on their head. This is because humans have < 1, 000, 000 hairs and there are > 1, 000, 000 people in NYC. The pigeonhole principle is particularly powerful in existence proofs which are not constructive. For example in the previous example we proved the existence of two ...
... have the same number of hairs on their head. This is because humans have < 1, 000, 000 hairs and there are > 1, 000, 000 people in NYC. The pigeonhole principle is particularly powerful in existence proofs which are not constructive. For example in the previous example we proved the existence of two ...
lect13 - Kent State University
... • Cantor proposed a rather nice solution to this problem. He observed that two finite sets have the same size if the elements of one set can be paired with the elements of the other set. This idea can be extended to infinite sets. • Assume we have two sets A and B and a function f from A to B. • Say ...
... • Cantor proposed a rather nice solution to this problem. He observed that two finite sets have the same size if the elements of one set can be paired with the elements of the other set. This idea can be extended to infinite sets. • Assume we have two sets A and B and a function f from A to B. • Say ...
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... Corollary: Arithmetic is not axiomatizable. Gödel’s incompleteness theorem is often described as “any consistent and sufficiently strong formal theory of arithmetic is incomplete”, where a formal theory is viewed as one whose theorems are derivable from an axiom system. For such theories there will ...
... Corollary: Arithmetic is not axiomatizable. Gödel’s incompleteness theorem is often described as “any consistent and sufficiently strong formal theory of arithmetic is incomplete”, where a formal theory is viewed as one whose theorems are derivable from an axiom system. For such theories there will ...
Truth, Conservativeness and Provability
... important requirement for deflationary truth theories. It is the deflationist’s intuition that truth is in some sense ‘innocent’ or ‘metaphysically thin’. 1 The truth predicate is just a ‘logical device’ permitting us to formulate useful generalizations (moreover, some of these generalizations will ...
... important requirement for deflationary truth theories. It is the deflationist’s intuition that truth is in some sense ‘innocent’ or ‘metaphysically thin’. 1 The truth predicate is just a ‘logical device’ permitting us to formulate useful generalizations (moreover, some of these generalizations will ...
Finite and Infinite Sets. Countability. Proof Techniques
... We count first all elements with index 1, then we count all elements with index 2, etc. Important: it is possible to count all the first elements (and then the second, third, etc) because we have only 3 sets: A, B, C, i.e. finite number of sets, so we can put the sets themselves (not the elements of ...
... We count first all elements with index 1, then we count all elements with index 2, etc. Important: it is possible to count all the first elements (and then the second, third, etc) because we have only 3 sets: A, B, C, i.e. finite number of sets, so we can put the sets themselves (not the elements of ...
INTRODUCTION TO GROUP THEORY (MATH 10005) The main
... Often, especially when we’re dealing with abstract properties of general groups, we’ll simplify the notation by writing xy instead of x?y, as though we’re multiplying. In this case we’ll say, for example, “Let G be a multiplicatively-written group”. Note that this is purely a matter of the notation ...
... Often, especially when we’re dealing with abstract properties of general groups, we’ll simplify the notation by writing xy instead of x?y, as though we’re multiplying. In this case we’ll say, for example, “Let G be a multiplicatively-written group”. Note that this is purely a matter of the notation ...
Hilbert Calculus
... Using S ∪ {F } ⊢ F and Modus Ponens we get S ∪ {F } ⊢ G. Assume S ∪ {F } ⊢ G. Proof by induction on the derivation (length): Axiom/Hypothesis: G is instance of an axiom or G ∈ S ∪ {F }. If F = G use example of derivation to prove S ⊢ F → F . Otherwise S ⊢ G and by Axiom (1) S ⊢ G → (F → G). By Modus ...
... Using S ∪ {F } ⊢ F and Modus Ponens we get S ∪ {F } ⊢ G. Assume S ∪ {F } ⊢ G. Proof by induction on the derivation (length): Axiom/Hypothesis: G is instance of an axiom or G ∈ S ∪ {F }. If F = G use example of derivation to prove S ⊢ F → F . Otherwise S ⊢ G and by Axiom (1) S ⊢ G → (F → G). By Modus ...
