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Enumerations in computable structure theory
Enumerations in computable structure theory

... is Σ0α (A) if ϕ(x) is computable Σα , and Π0α (A) if ϕ(x) is computable Πα . Moreover, this holds with all imaginable uniformity, over structures and formulas. It is easy to see that if A has a formally c.e. Scott family, then it is relatively computably categorical, so it is computably categorical ...
Enumerations in computable structure theory
Enumerations in computable structure theory

... the same–then we could drop the effectiveness conditions from Goncharov’s result. However, Goncharov [11] showed that this is not the case, using an enumeration result of Selivanov [17]. There are examples with further properties. Cholak, Goncharov, Khoussainov, and Shore [9] gave an example of a str ...
Barwise: Infinitary Logic and Admissible Sets
Barwise: Infinitary Logic and Admissible Sets

... We say that two structures M and N , of arbitrary cardinality, are potentially isomorphic if there is a back-and-forth family for M, N . It is obvious that isomorphic structures are potentially isomorphic. In the other direction, potentially isomorphic structures are very similar to each other, but ...
Proof, Sets, and Logic - Boise State University
Proof, Sets, and Logic - Boise State University

... press because it prevents the truncation of the construction of the cumulative hierarchy of isomorphism types of well-founded extensional relations (the world of the usual set theory). Further, it is interesting to note that the von Neumann definition of ordinal numbers is entirely natural if the wo ...
Proof, Sets, and Logic - Department of Mathematics
Proof, Sets, and Logic - Department of Mathematics

... the i operation. It is certainly a valid point which is being made, but I would like to say it neatly. It is nice to get to it before one tries to construct NF(U); thus one does not naturally wander into the NF problem. July 9, 2009: I am editing the text again, much later, with an eye to starting t ...
this PDF file
this PDF file

... Now we come to the important question: what is a sufficient condition on the set S for UPFS to hold? From Theorems 4 and 5, we have two necessary conditions for UPF-S to hold: (C1) S is infinite and (C2) S contains all primes. By ET, we know that the condition (C2) is stronger than (C1). But let us pre ...
possible-worlds semantics for modal notions conceived as predicates
possible-worlds semantics for modal notions conceived as predicates

Lectures on Proof Theory - Create and Use Your home.uchicago
Lectures on Proof Theory - Create and Use Your home.uchicago

... consequence of the following: ...
Theories of arithmetics in finite models
Theories of arithmetics in finite models

... of addition and multiplication with the concatenation operation. We consider also the spectrum problem. We show that the spectrum of arithmetic with multiplication and arithmetic with exponentiation is strictly contained in the spectrum of arithmetic with addition and multiplication. ...
Internal Inconsistency and the Reform of Naïve Set Comprehension
Internal Inconsistency and the Reform of Naïve Set Comprehension

... numbers. The next highest ordinal is itself an ordinal (because of its own description) and (by definition based on the concept of the ordering of the series of ordinals) larger not only than all ordinals including its immediate predecessor (the set of all ordinals) but also than itself. In this cas ...
Contents 1 The Natural Numbers
Contents 1 The Natural Numbers

... k(0) = a and k(n0 ) = hn (k(n)) = ak(n). This is precisely the exponential function f of the first example. For the second example, let A = N and now choose a = 1. Define hn by the formula hn (x) = n0 x. The reader should convince himeself/herself that this definition does not use induction. In this ...
Recurrent points and hyperarithmetic sets
Recurrent points and hyperarithmetic sets

... The minimal sets are pairwise disjoint closed sets, which might, but need not, partition the set of recurrent points. Here are some examples. 1·7 EXAMPLE Let X = [0, 1] and f (x) ≡ x2 : then A∞ = {0, 1}, although f is onto and 1-1 and X is compact. The minimal sets in this case are {0} and {1}. 1·8 ...
Mathematical Induction
Mathematical Induction

... Induction – a powerful method of proof P(1) x (P(x)P(x+1)) ------------------------------------x P(x) ...
Proof Theory: From Arithmetic to Set Theory
Proof Theory: From Arithmetic to Set Theory

... • “No one shall drive us from the paradise which Cantor has created for us.” ...
The Model-Theoretic Ordinal Analysis of Theories of Predicative
The Model-Theoretic Ordinal Analysis of Theories of Predicative

... though they differ from the "standard"assignments only slightly. Note that different notations can denote the same ordinal, as is the case with so and cowo.Further note that equivalent notations need not have equivalent limit sequences; for example, eo[n] = On+2, but co60[n] = e)'n+2+l. We assume th ...
Translating the Hypergame Paradox - UvA-DARE
Translating the Hypergame Paradox - UvA-DARE

... Note that here the argument which leads to the paradox has an asymmetric pattern which consists of two parts: (1) ‘p is true’ (hypergame is a founded game); (2) ‘if p is true then lp is true’ (if hypergame is founded, then it is not founded). In other words, we arrive to a contradiction by showing ‘ ...
Situation 39: Summing Natural Numbers
Situation 39: Summing Natural Numbers

... The set of integers from 1 to n is an ordered sequence of natural numbers. Symmetry and the ordered nature of the sequence allow for rearrangements of a particular kind that facilitate finding a sum; under any such rearrangement of the elements of a discrete set, the cardinality remains the same, as ...
On the regular extension axiom and its variants
On the regular extension axiom and its variants

... Proof: Every set x is contained in a transitive set A with ω ⊆ A. Thus if we can show that H(A) is a set we have found a set comprising x which is functionally regular. The main task of the proof is therefore to show that H(A) is a set. Let ρ be the supremum of all ordinals which are order types of ...
CARLOS AUGUSTO DI PRISCO The notion of infinite appears in
CARLOS AUGUSTO DI PRISCO The notion of infinite appears in

... The construction of the model M [g] and the proof that it has the desired properties is quite elaborate. Certain elements of the model M are used as “names” for elements of M [g]. Which set is the object named by a name τ depends on the generic g, and given g, the model M [g] is the collection of se ...
Chapter 4, Mathematics
Chapter 4, Mathematics

... Just defining addition and multiplication like this is not sufficient to establish integer arithmetic. We need to show also that the definitions are consistent and correspond to the operations of addition and multiplication for integers. We have defined integers as equivalence classes of pairs of na ...
Revised Version 080113
Revised Version 080113

... A symbolic manipulation of the formula for the sum of the first n natural € in the appendix. numbers using even and odd numbers for n can be found Mathematical Focus 2 Specific examples suggest a general formula for the sum of the first n natural numbers. Strategic choices for pair-wise grouping of ...
HERE
HERE

... set, the cardinality remains the same, as does the sum of the elements. A formula ...
Goldbach`s Conjecture
Goldbach`s Conjecture

... Unfortunatly, this relationship does not prove Goldbach’s Conjecture, it only shows that any ‘disproofs’ will likely be isolated cases. Research is continuing towards the following: Theorem 4. There exists no n such that φC (n + m) ≥ π(n + m) for all m ∈ N. This appears to be true because we can con ...
On interpretations of arithmetic and set theory
On interpretations of arithmetic and set theory

... L -theory is a consistent set of L -sentences (not necessarily complete or closed under deduction) and we make the convenient assumptions that all languages are purely relational, equality (=) being one of the relation symbols, and a further unary relation Dom() for the domain is always present. Ful ...
natural numbers
natural numbers

... •the set of positive integers {1, 2, 3, ...} according to the traditional definition, or •the set of non-negative integers {0, 1, 2, ...} according to a definition first appearing in the nineteenth century The have two main purposes: •counting, and •Ordering Properties of the natural numbers are stu ...
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Ordinal arithmetic

In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set which represents the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. The so-called ""natural"" arithmetical operations retain commutativity at the expense of continuity.
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