July | 2007 | A kind of library

Goodstein sequences

July 27, 2007

Will Sladek, a student at Caltech, wrote an excellent introductory paper on incompleteness in $\mathsf{PA}$ , The termite and the tower. While Will was working on his paper, I wrote a short note, Goodstein’s function, on how to compute Goodstein’s function. Please let me know of any comments of corrections to either article.

Update: My paper has been published.

Goodstein’s function. Revista Colombiana de Matemáticas 41 (2)(2007), 381–391. MR2585906 (2011c:03139). Zbl 1156.03053.

2 Comments | Incompleteness | Tagged: Goodstein's function, Undecidability and incompleteness | Permalink
Posted by andrescaicedo

You are currently browsing the A kind of library blog archives for July, 2007.
Categories

Categories
Search

Mathematical links
My pages
Others
Mathoverflow
- Answer by Andrés E. Caicedo for Riemann series theorem and uncountable number of sums which sum to every value November 1, 2019
  Every $P_c$ has the size of the reals. For instance, suppose $\sum_n a_n=c$ and start by writing $\mathbb N=A\cup B$ where $\sum_{n\in A}a_n$ converges absolutely (to $a$, say). This is possible because $a_n\to 0$: Let $m_0
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for "Inner Regularity" of probability measure on totally ordered sets October 7, 2019
  Consider a subset $\Omega$ of $\mathbb R$ of size $\aleph_1$ and ordered in type $\omega_1$. (This uses the axiom of choice.) Let $\mathcal F$ be the $\sigma$-algebra generated by the initial segments of $\Omega$ under the well-ordering (so all sets in $\mathcal F$ are countable or co-countable), with the measure that assigns $0$ to the countable sets and $1 […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Question about Jech's proof of V = L implies GCH June 3, 2019
  You assume $\omega_\alpha\subseteq M$ and $X\in M$ so that $X$ belongs to the transitive collapse of $M$ (because if $\pi$ is the collapsing map, $\pi(X)=\pi[X]=X$. You assume $|M|=\aleph_\alpha$ so that the transitive collapse of $M$ has size $\aleph_\alpha$. Since you also have that this transitive collapse is of the form $L_\beta$ for some $\beta$, it fol […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Smallest $\beta$ such that it is provable that $2^{\aleph_\beta} > 2^{\aleph_0}$ January 16, 2019
  Perhaps the following may clarify the comments: for any ordinal $\delta$, there is a Boolean-valued extension of the universe of sets where $2^{\aleph_0}>\aleph_\delta$ holds. If you rather talk of models than Boolean-valued extensions, what this says is that we can force while preserving all ordinals, and in fact all initial ordinals, and make the contin […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Arguments against large cardinals October 29, 2010
  I do not know of any active set theorists who think large cardinals are inconsistent. At least, within the realm of cardinals we have seriously studied. [Reinhardt suggested an ultimate axiom of the form "there is a non-trivial elementary embedding $j:V\to V$". Though some serious set theorists found it of possible interest immediately following it […]
  Andrés E. Caicedo
Math.StackExchange
- Answer by Andrés E. Caicedo for How many integer solutions are there to $x+y+z=8$ February 12, 2020
  For the second problem, write $x=-3+x'$ and so on. You have $x'+y'+z'=17$ and $x',\dots$ are nonnegative, a case you know how to solve. You can also solve the first problem this way; now you would set $x=1+x'$, etc.
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Cofinality of $2^{ February 10, 2020
  I assume you want to require that $\kappa$ is a limit cardinal; otherwise, if $\kappa=\rho^+$ then $2^{
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for What does it mean when we say that stationary sets are analogous to sets of non-zero measure? February 7, 2020
  Given a complete measure space $X$, consider the collection $F$ of full measure sets. These are the sets whose complement has measure 0. Note that if $Y\in F$ then any superset of $Y$ (that is, any $Z$ with $Y\subseteq Z\subseteq X$) is in $F$ as well. (This is why I require the measure to be complete, which means that any subset of a measure zero set is mea […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Help with proof of existence of $G_\delta$ sets $G$ such that both $G$ and its complement have uncountable intersection with any open interval January 30, 2020
  Although the $G_j$ are decreasing, their intersection is far from just being the rationals. As indicated in the original post, in fact $G$ meets each open interval in uncountably many points. A nowhere dense subset of $\mathbb R$ is a set $X\subseteq \mathbb R$ with the property that any nonempty open set $U\subseteq\mathbb R$ contains a nonempty open subset […]
  Andrés E. Caicedo
- Answer by Andrés E. Caicedo for Countable and uncountable number December 12, 2019
  A set $G\subseteq \mathbb R$ is $G_\delta$ (that is, a countable intersection of open sets) if and only if there exists a function $f\!:\mathbb R\to\mathbb R$ that is continuous at each point of $G$ and discontinuous at each point of its complement. The result you are requesting follows from noticing that there are $G_\delta$ sets $G$ such that both $G$ and […]
  Andrés E. Caicedo
July 2007

M T W T F S S

« Mar Sep »

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 31
Archives
- June 2019 (2)
- February 2019 (7)
- January 2019 (1)
- December 2018 (1)
- October 2018 (2)
- August 2018 (1)
- 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 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 Paul Larson 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

	Anthea M. on No durians
	580 -Some choiceless… on 580 -Some choiceless results…
	Set theory seminar -… on Set theory seminar -Forcing ax…
	Set theory seminar -… on Set theory seminar -Forcing ax…
	Set theory seminar -… on Set theory seminar -Forcing ax…
	Set theory seminar -… on Set theory seminar -Forcing ax…
	Set theory seminar -… on Set theory seminar -Forcing ax…
	Set theory seminar -… on Set theory seminar -Forcing ax…
	Set theory seminar -… on Set theory seminar -Forcing ax…
	Set theory seminar -… on Set theory seminar -Forcing ax…
	Set theory seminar -… on Set theory seminar -Forcing ax…
	Set theory seminar -… on Set theory seminar -Forcing ax…
	Set theory seminar -… on Set theory seminar -Forcing ax…
	Set theory seminar -… on Set theory seminar -Forcing ax…
	Set theory seminar -… on Set theory seminar -Forcing ax…

July 2007
M	T	W	T	F	S	S
« Mar				Sep »
						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	31

A kind of library

Goodstein sequences

Categories

Recent Comments

Mathematical links

My pages

Others

Mathoverflow

Math.StackExchange

Archives