completeness | A kind of library

502 – Completeness

September 25, 2009

1. Preliminaries

Just as with propositional logic, one can define a proof system for first order logic. We first need two definitions.

Definition 1 If ${\phi}$ is a formula, a generalization of ${\phi}$ is any formula of the form

$\displaystyle \forall x_1\forall x_2\dots\forall x_n\,\phi.$

(We are not requiring in the definition above that the ${x_i}$ are different or that they are present in ${\phi.}$ )

Definition 2 Let ${t}$ be a term, let ${x}$ be a variable, and let ${\phi}$ be a formula. We say that ${t}$ is substitutable for ${x}$ in ${\phi}$ iff no free occurrence of ${x}$ in ${\phi}$ is within the scope of a quantifier ${\exists z}$ or ${\forall z}$ with ${z}$ a variable occurring in ${t.}$

Read the rest of this entry »

We now want to show that whenever ${S\models A,}$ then also ${S\vdash A.}$ Combined with the soundness Theorem 22, this shows that the notions of provable and true coincide for propositional logic, just as they did for the tree system. The examples above should hint at how flexible and useful this result actually is. This will be even more evident for first order (predicate) logic.

Theorem 26 (Completeness) For any theory ${S}$ and any formula ${A,}$ if ${S\models A,}$ then ${S\vdash A.}$

Read the rest of this entry »

Leave a Comment » | 502: Logic and set theory | Tagged: completeness, propositional logic | Permalink
Posted by andrescaicedo

502 – Formal systems (3)

September 4, 2009

Theorem 2 (Completeness) Whenever ${\Sigma\models\tau,}$ then also ${\Sigma\vdash\tau.}$

Read the rest of this entry »

Leave a Comment » | 502: Logic and set theory | Tagged: completeness | Permalink
Posted by andrescaicedo

Categories

Categories
Search

Mathematical links
My pages
Others
Mathoverflow
- Relative Densities September 12, 2013
  Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory. How can I count the relative density of those $p$ (of the form $3k+1$) such that the equation $2=3x^3$ has no solutions modulo $p$? Really, even pointers on how to say anything […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Solutions to the Continuum Hypothesis May 18, 2010
  (1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Are there any non-linear solutions of Cauchy's equation ($f(x+y)=f(x)+f(y)$) without assuming the Axiom of Choice? March 6, 2011
  It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here (Intern […]
  Andrés E. Caicedo
- David Gale's subset take-away game October 12, 2010
  I learned of this problem through Su Gao, who heard of it years ago while a post-doc at Caltech. David Gale introduced this game in the 70s, I believe. I am only aware of two references in print: Richard K. Guy. Unsolved problems in combinatorial games. In Games of No Chance, (R. J. Nowakowski ed.) MSRI Publications 29, Cambridge University Press, 1996, pp. […]
  Andrés E. Caicedo
- A translation of the Cantor set contained in the irrationals February 22, 2014
  Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]
  Andrés E. Caicedo
Math.StackExchange
- Answer by Andrés E. Caicedo for Well-ordering of the reals in ZF with constructibility? April 19, 2018
  $L$ has such a nice canonical structure that one can use it to define a global well-ordering. That is, there is a formula $\phi(u,v)$ that (provably in $\mathsf{ZFC}$) well-orders all of $L$, so that its restriction to any specific set $A$ is a set well-ordering of $A$. The well-ordering $\varphi$ you are asking about can be obtained as the restriction to $\ […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Assuming ZF and the axiom of constructibility, $\alpha > \omega, V_\alpha = L_\alpha \iff \alpha = \aleph_\alpha$ February 28, 2018
  Define the sequence of beth numbers by $\beth_0=\aleph_0$, $\beth_{\alpha+1}=2^{\beth_\alpha}$ and $\beth_\lambda=\sup_{\beta
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for On the definition of Inductive set. Equivalences February 18, 2018
  The two concepts are different. For example, $\omega$, the first infinite ordinal, is the standard example of an inductive set according to the first definition, but is not inductive in the second sense. In fact, no set can be inductive in both senses (any such putative set would contain all ordinals). In the context of set theory, the usual use of the term […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Find a contradiction to: If $t^n < (\sup \{t \in \mathbb{R}_{>0}: t^n < x\})^n$ then $t^n < x$ when this supremum is greater than $x$. February 16, 2018
  I will show that for any positive integers $n,\ell,k$ there is an $M$ so large that for all positive integers $i$, if $i/M\le \ell$, then the difference $$ \left(\frac iM\right)^n-\left(\frac{i-1}M\right)^n $$ is less than $1/k$. Let's prove this first, and then argue that the result follows from it. Note that $$ (i+1)^n-i^n=\sum_{k=0}^{n-1}\binom nk i^ […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Prove by induction that $n^2>7n+1$ for all $n \ge 8$ February 15, 2018
  I think it is cleaner to argue without induction. If $n$ is a positive integer and $n\ge 8$, then $7n$ is both less than $n^2$ and a multiple of $n$, so at most $n^2-n$ and therefore $7n+1$ is at most $n^2-n+1
  Andrés E. Caicedo
April 2018

M T W T F S S

« Feb

1

2 3 4 5 6 7 8

9 10 11 12 13 14 15

16 17 18 19 20 21 22

23 24 25 26 27 28 29

30
Archives
- February 2018 (1)
- October 2017 (1)
- July 2017 (1)
- September 2016 (2)
- August 2016 (3)
- July 2016 (2)
- April 2016 (1)
- October 2015 (1)
- June 2015 (1)
- May 2015 (1)
- April 2015 (7)
- March 2015 (5)
- February 2015 (7)
- January 2015 (11)
- December 2014 (2)
- November 2014 (5)
- October 2014 (6)
- September 2014 (3)
- August 2014 (3)
- April 2014 (3)
- March 2014 (5)
- February 2014 (2)
- January 2014 (2)
- December 2013 (2)
- November 2013 (10)
- October 2013 (6)
- September 2013 (7)
- August 2013 (5)
- June 2013 (1)
- May 2013 (8)
- April 2013 (7)
- March 2013 (5)
- February 2013 (9)
- January 2013 (2)
- November 2012 (1)
- September 2012 (1)
- August 2012 (3)
- May 2012 (3)
- April 2012 (10)
- March 2012 (4)
- February 2012 (13)
- January 2012 (15)
- November 2011 (8)
- October 2011 (14)
- September 2011 (7)
- August 2011 (1)
- April 2011 (1)
- March 2011 (3)
- February 2011 (4)
- January 2011 (7)
- December 2010 (1)
- November 2010 (7)
- October 2010 (12)
- September 2010 (13)
- August 2010 (3)
- May 2010 (4)
- April 2010 (11)
- March 2010 (6)
- February 2010 (12)
- December 2009 (9)
- November 2009 (11)
- October 2009 (13)
- September 2009 (19)
- August 2009 (10)
- June 2009 (1)
- May 2009 (2)
- April 2009 (14)
- March 2009 (17)
- February 2009 (19)
- January 2009 (15)
- December 2008 (5)
- November 2008 (12)
- October 2008 (17)
- September 2008 (10)
- August 2008 (3)
- June 2008 (3)
- May 2008 (15)
- April 2008 (21)
- March 2008 (10)
- February 2008 (13)
- January 2008 (11)
- September 2007 (1)
- July 2007 (1)
- March 2007 (4)
- February 2007 (20)
- January 2007 (23)
- December 2006 (4)
Alan Turing Alexander S. Kechris Alfred Tarski algebraic numbers Andras Hajnal Andreas Blass Andrew Michael Bruckner arithmetic mean-geometric mean inequality Arnoud C. M. van Rooij Augustin Cauchy Axel Thue Bernhard Riemann BEST bpfa Bukovsky-Hechler theorem compactness completeness covering property determinacy Donald A. Martin elementary embedding Emile Borel Euclidean algorithm extension field field field extension Francisco Frank Plumpton Ramsey Fred Galvin Galvin-Hajnal theorem GCH Georg Cantor Godfrey Harold Hardy Grigor Sargsyan Harvey Friedman ideal Isaac Newton James Baumgartner Jerome Keisler Johan Thim John R. Steel Karl Theodor Wilhelm Weierstrass Koenig's lemma Kurt Goedel Lech Maligranda Marion Scheepers measurable cardinal Menachem Magidor midterm mm Nowhere differentiable function Paul Erdos pfa propositional logic quartic equation Ramsey theorem Rene Baire Richard Rado Robert Solovay Saharon Shelah Samuel Coskey sch Silver's theorem Stefan Banach Stevo Todorcevic supercompact cardinals Tarski's conjecture Turing degrees ultrafilter ultrapower Undecidability and incompleteness Unsolvable problems W. Hugh Woodin Wacław Franciszek Sierpiński Wilhelmus Hendricus Schikhof

	116c- Homework 7 \| A… on 116c- Lecture 18
	116c- Lecture 17 \| A… on 116c: Mathematical Logic (Set…
	116c- Lecture 18 \| A… on 116c- Homework 7
	116c- Lecture 19 \| A… on 116c- Lecture 13
	580 -Partition calcu… on 580 -III. Partition calcu…
	580 -Partition calcu… on 580 -Partition calculus (…
	Nosferatu \| A kind o… on The cabinet of Doctor Cal…
	Monica Agana –… on 414/514 The theorems of Rieman…
	The Dalek invasion o… on The Dalek invasion of Boi…
	andrescaicedo on Help!
	Asaf Karagila on On anti-foundation and coding…
	andrescaicedo on On anti-foundation and coding…
	Asaf Karagila on On anti-foundation and coding…
	On anti-foundation a… on Interpretations
	andrescaicedo on MSC2020

A kind of library

502 – Completeness

502 – Propositional logic (4)

502 – Formal systems (3)

Categories

Recent Comments

Mathematical links

My pages

Others

Mathoverflow

Math.StackExchange

Archives

April 2018
M	T	W	T	F	S	S
« Feb
						1
2	3	4	5	6	7	8
9	10	11	12	13	14	15
16	17	18	19	20	21	22
23	24	25	26	27	28	29
30