Completeness of the real numbers
... impossible, since it would imply a > L, an upper bound for A. Thus a0 ∈ A is an upper bound for A, and hence is the largest element of A. In particular, if b ∈ R and b < a0 , b can’t be an upper bound for A. Thus a0 = supR (A), so a0 = L. It then follows (see handout 6) that R has the ‘density prope ...
... impossible, since it would imply a > L, an upper bound for A. Thus a0 ∈ A is an upper bound for A, and hence is the largest element of A. In particular, if b ∈ R and b < a0 , b can’t be an upper bound for A. Thus a0 = supR (A), so a0 = L. It then follows (see handout 6) that R has the ‘density prope ...
1 Introduction to Logic
... the one at right. Assume an unlimited supply of 5- and 7gram weights. To weigh a potato, we put the potato in one pan of the balance scale and add weights to one or both pans until the scale balances. The question is: What sizes of potatoes can we weigh with our apparatus and weights? Using only the ...
... the one at right. Assume an unlimited supply of 5- and 7gram weights. To weigh a potato, we put the potato in one pan of the balance scale and add weights to one or both pans until the scale balances. The question is: What sizes of potatoes can we weigh with our apparatus and weights? Using only the ...
SOME REMARKS ON SET THEORY, IX. COMBINATORIAL
... the example just constructed, m > 1/r - E . We do not know how far this inequality can be improved ; we only have some special results which show that if m is sufficiently large for a set-function F on S, of type 2 and of order greater than 1 - 1/r, then there always exists a complete (r + 1)-gon wi ...
... the example just constructed, m > 1/r - E . We do not know how far this inequality can be improved ; we only have some special results which show that if m is sufficiently large for a set-function F on S, of type 2 and of order greater than 1 - 1/r, then there always exists a complete (r + 1)-gon wi ...
A NEW STRONG INVARIANCE PRINCIPLE FOR SUMS OF
... or, equivalently, E[|X| /(LL|X|)p ] < ∞. In the finite dimensional case this condition follows from (b). This was already pointed out in the 1-dimensional case (see, for instance, [6]), and we shall give here a detailed proof of this fact in arbitrary finite dimension. We mention that this implicati ...
... or, equivalently, E[|X| /(LL|X|)p ] < ∞. In the finite dimensional case this condition follows from (b). This was already pointed out in the 1-dimensional case (see, for instance, [6]), and we shall give here a detailed proof of this fact in arbitrary finite dimension. We mention that this implicati ...
Gödel`s correspondence on proof theory and constructive mathematics
... etc. I will do something less synoptic but perhaps complementary to Mancosu’s essay: I will concentrate more on small details in the correspondence that seem to me worthy of further discussion; although there are many of these and I will not take up all or even most of them. In fact, I will restric ...
... etc. I will do something less synoptic but perhaps complementary to Mancosu’s essay: I will concentrate more on small details in the correspondence that seem to me worthy of further discussion; although there are many of these and I will not take up all or even most of them. In fact, I will restric ...
Pythagoras and the Pythagoreans
... a single number, the Unit, (i.e. one). They treated the unit, which is a point without position, as a point, and a point as a unit having position. The unit was not originally considered a number, because a measure is not the things measured, but the measure of the One is the beginning of number.6 T ...
... a single number, the Unit, (i.e. one). They treated the unit, which is a point without position, as a point, and a point as a unit having position. The unit was not originally considered a number, because a measure is not the things measured, but the measure of the One is the beginning of number.6 T ...
Advanced Topics in Propositional Logic
... Go through these, and whenever you encounter Ai such that neither it nor its negation is derivable from , add Ai to . In view of Lemma 5, you will end up with a formally complete set. To see that the same set is also formally consistent, suppose, for a contradiction, that it is not. Consider the s ...
... Go through these, and whenever you encounter Ai such that neither it nor its negation is derivable from , add Ai to . In view of Lemma 5, you will end up with a formally complete set. To see that the same set is also formally consistent, suppose, for a contradiction, that it is not. Consider the s ...
Generalized Cantor Expansions - Rose
... life. A less common way to represent a number is the so called Cantor expansion. Often presented as exercises in discrete math and computer science courses [8.2, 8.5], this system uses factorials rather than exponentials as the basis for the representation. It can be shown that the expansion is uniq ...
... life. A less common way to represent a number is the so called Cantor expansion. Often presented as exercises in discrete math and computer science courses [8.2, 8.5], this system uses factorials rather than exponentials as the basis for the representation. It can be shown that the expansion is uniq ...
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.