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Topic: List of statements undecidable in ZFC

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2003

	Whitehead problem Information
		Since the consistency of ZFC implies both the consistency of the axiom that all sets are constructible, and the consistency of Martin's axiom plus the negation of the continuum hypothesis, this shows that Whitehead's problem is undecidable.
		While the existence of undecidable statements had been known since Gödel's incompleteness theorem of 1931, previous examples of undecidable statements (such as the continuum hypothesis) had been confined to the realm of set theory.
		Various similar independence statements were proved and it was realized more and more that the theory of uncountable abelian groups depends very sensitively on the underlying set theory.
	www.bookrags.com /wiki/Whitehead_problem (376 words)

	Lists for mathematics - Eua4xiacwiki
		List of harmonic analysis and representation theory topics
		List of terms relating to algorithms and data structures
		List of it.wp liste di matematici in Italian
	alice.iac.rm.cnr.it:8080 /wiki/index.php/Lists_for_mathematics (131 words)

	List of lists of mathematical topics (Site not responding. Last check: 2007-11-06)
		List of harmonic analysis and representation theory topics
		List of letters used in mathematics and science
		List of terms relating to algorithms and data structures
	hallencyclopedia.com /List_of_lists_of_mathematical_topics (391 words)

	List of statements undecidable in ZFC -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-06)
		The following is a list of (additional info and facts about mathematical) mathematical statements that are undecidable in (additional info and facts about ZFC) ZFC (the Zermelo-Fraenkel axioms plus the (additional info and facts about axiom of choice) axiom of choice), assuming that ZFC is consistent.
		Charles Akemann and Nik Weaver showed in 2003 that the statement "there exists a counterexample to (additional info and facts about Naimark's problem) Naimark's problem which is generated by (additional info and facts about ℵ) ℵ
		for every n) are independent of ZFC (as shown by (additional info and facts about Paul Cohen) Paul Cohen and (additional info and facts about Kurt Gödel) Kurt Gödel), as is the combinatorial statement (additional info and facts about ◊) ◊ (which implies CH).
	www.absoluteastronomy.com /encyclopedia/l/li/list_of_statements_undecidable_in_zfc2.htm (289 words)

	List of statements undecidable in ZFC (Site not responding. Last check: 2007-11-06)
		__NOTOC__ The following is a list of mathematical statements that are undecidable in ZFC (the Zermelo-Fraenkel axioms plus the axiom of choice), assuming that ZFC is consistent.
		Martin's Axiom implies that there exists a function on the unit square whose iterated integrals are not equal, while as a variant of Freiling's Axiom of Symmetry implies that in fact a strong Fubini type theorem for [0, 1] does hold, and whenever the two iterated integrals exist they are equal.
		There are many cardinal invariants of the real line, connected with measure theory and statements related to the Baire category theorem whose exact values are independent of ZFC (in a stronger sense than that the continuum hypothesis is in ZFC.
	www.worldhistory.com /wiki/L/List-of-statements-undecidable-in-ZFC.htm (451 words)

	Discover the Wisdom of Mankind on Blinkbits.com
		List of state-named roadways in Washington, D.C. (en)
		List of states of Venezuela by population (en)
		List of states where language is a political issue (en)
	www.blinkbits.com /wikifeeds/LI?from=91500 (1043 words)

	Infinite Ink: The Continuum Hypothesis FAQ
		This problem was so important that Hilbert put it first in his list of 23 problems that he thought were the most important for twentieth century mathematics.
		This means that there are statements in the language of set theory -- called undecidable statements -- that can neither be proved nor disproved.
		Mathematicians suspected that CH was undecidable in ZFC but it took until 1963 until this was proved.
	www.ii.com /math/ch/faq (520 words)

	From Frege To Godel: von Heijenoort (Site not responding. Last check: 2007-11-06)
		Richard's paradox results from listing "all numbers that are defined by finitely many words" and performing a diagonalization to construct a number, via finitely many words, which cannot be in the list.
		To prohibit existence statements and the principle of excluded middle is tantamount to relinquishing the science of mathematics altogether.
		The argument is based on the Liar paradox, except instead of considering a statement expressing "I am not true," he considers a statement expressing "I am not provable." The diagonalization used to construct such a statement is reminiscent of Cantor's diagonal procedure and Richard's paradox.
	www.andrew.cmu.edu /user/cebrown/notes/vonHeijenoort.html (8419 words)

	Afamcinema DVD: Godel's Theorem: An Incomplete Guide to Its Use and Abuse - $23.70 (Site not responding. Last check: 2007-11-06)
		As noted previously, Goedel's alleged proof of this statement is said dеrivеd from outside the system and accordingly not to be contradicting the statement.
		What is significant is that the statement, thus harboring contradictions, cannot be added to the axioms of thе system as suggested by the discussants, because that would make thе system inconsistent, with consistency vehemently, and justly, insisted on by all authors.
		However, it should bе noted that Franzen's analogy to the systems РА and ZFC regarding self-understanding is itself nothing more than an analogy, and thus suffers from the same problems as thе analogy of Hofstadter's regarding the inability to attain self-understanding.
	www.afamcinema.com /good31353638383132333838.html (2545 words)

	Infinite Ink: The Continuum Hypothesis FAQ
		This problem was so important that Hilbert put it first in his list of 23 problems that he thought were the most important for twentieth century mathematics.
		This means that there are statements in the language of set theory -- called undecidable statements -- that can neither be proved nor disproved.
		Mathematicians suspected that CH was undecidable in ZFC but it took until 1963 until this was proved.
	ii.best.vwh.net /math/ch/faq (520 words)

	sciforums.com - A Godel Question
		It is actually quite easy to construct "axiom systems" in which any statement can be either proved or disproved (many "college geometry" texts include examples of finite geometries in which the axioms assert that "there exist exactly 3 points" in which every statement can be either proved or disproved).
		Also the undecidable statements of Goedel seem to be interpreted as logical contradictions by some.
		an undecidable statement is one that is true, but does not follow from the axioms.
	www.sciforums.com /showthread.php?t=19617 (2516 words)

	GuruNet — Content Map
		List of state Republican Parties in the U.S. List of state route markers
		List of Stewards of the Manor of Northstead
		List of Stewards of the Manor of Poynings
	www.gurunet.com /cm-dsid-2222-letter-1L-first-27851 (109 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		It then comes as somewhat of a surprise to the student and novice mathematician that there are statements in mathematics that are neither true nor false, questions to which the rules of the game do not provide a specific answer.
		The fact that CH is and ~CH have been shown to be consistent with ZFC implies either that CH is independent of ZFC or that ZFC is inconsistent and thus flawed.
		The Continuum Hypothesis would be the first problem enumerated by Hilbert at the turn of the 20th century in his list of 23 unsolved problems.
	mars.drw.net /acw83/draft.doc (728 words)

	ZFC - OneLook Dictionary Search
		ZFC : Stammtisch Beau Fleuve Acronyms [home, info]
		ZFC : Free On-line Dictionary of Computing [home, info]
		Phrases that include ZFC: list of statements undecidable in zfc, zfc set, zfc set theory
	www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=ZFC (100 words)

	Antimeta: Truth Archives (Site not responding. Last check: 2007-11-06)
		This seems to be a largely accurate statement, though of course the existence of dozens and dozens (ok, maybe even hundreds) of mathematicians actively working in set theory, model theory, recursion theory, and proof theory shows that it's not totally true.
		Every statement a set theorist working in one structure makes can be translated into a statement about a generic extension of the universe by a set theorist working in the other.
		Thus, rather than "X was not brave" meaning that it is verifiable that X would have acted cowardly had she been exposed to danger, this anti-realist should say that "X was not brave" means that no process of inquiry will reveal that X would have acted bravely had she been exposed to danger.
	antimeta.org /blog/archives/truth (4387 words)

	[No title]
		It is recommended that each submission begin with a succinct statement of the problem, a summary of the main results, and a brief explanation of their significance, all suitable for a non-specialist.
		Organizers and publications, a list of authors referencing their talks and papers on the workshops, and a workshop bibliography are presented.
		There will be two lists: (1) one for complete semantics, and (2) the other for formal semantics of significant and large parts of languages, including information on which constructs are not covered.
	www.amast.org /archive/amast/links/v02/i01/AL0201.txt (9576 words)

	Antimeta: Truth Archives
		This seems to be a largely accurate statement, though of course the existence of dozens and dozens (ok, maybe even hundreds) of mathematicians actively working in set theory, model theory, recursion theory, and proof theory shows that it's not totally true.
		Every statement a set theorist working in one structure makes can be translated into a statement about a generic extension of the universe by a set theorist working in the other.
		Thus, rather than "X was not brave" meaning that it is verifiable that X would have acted cowardly had she been exposed to danger, this anti-realist should say that "X was not brave" means that no process of inquiry will reveal that X would have acted bravely had she been exposed to danger.
	www.ocf.berkeley.edu /~easwaran/blog/truth (7017 words)

	KH: Gödels Incompleteness Theorem
		In other words, the statement "there are no contradictions in the Principia system" cannot be proven true or false in the Principia system unless there are contradictions in the system (in which case it can be proven both true and false).
		However, the statement "We do not allow self-referential statements in Principia Mathematica" is a seeming violation of the rule against self-referential statements, an apparent contradiction at the heart of the philosophy, although it may be interpreted as meaning that none of the following statements in the formal system itself would be self-referential.
		That is, this statement may mean "in the following formal axiomatic system self-referential statements are not allowed," which clearly is not self-referential.
	www.hi.is /~joner/eaps/kh_Godel_11.htm (2735 words)

	Games Fresh : Article 'List of combinatorics topics' (Site not responding. Last check: 2007-11-06)
		A few decades ago it might have been said that combinatorics is to mathematics roughly what irritable bowel syndrome is to gastroenterology - a way to classify poorly-understood problems, and some standard remedies.
		This page is complementary to the list of graph theory topics: graph theory being the part of combinatorial mathematics that is most like a separate discipline.
		See also glossary of general topology for detailed definitions, the list of general topology topics and Main article.
	www.games-fresh.net /DisplayArticle285854.html (1163 words)

	AMIL-commentary
		In connection with conditional statements, as discussed at the bottom of page 21, here is a link to another discussion of the so-called material conditional (using the traditional horseshoe symbol for the conditional, instead of the arrow).
		An interesting "list of exercises" together with the "theory behind the exercises" be found (in pdf format) on a Stanford webpage.
		Then the undecidability of the theory of true arithmetic factors into two parts: For one, we will see that the theory of true arithmetic is not recursive.
	www.math.ucla.edu /~hbe/amil/commentary.html (9510 words)

	Hilbert's First and Second Problems and the foundations of mathematics by Peter J. Nyikos
		In 1900, David Hilbert gave a seminal lecture in which he spoke about a list of unsolved problems in mathematics that he deemed to be of outstanding importance.
		A set is of this cardinality if it is possible to list its members in an arrangement such that each one is encountered after a finite number (however large) of steps.
		Hilbert was already aware, at the time of his 1900 lecture, of some connection between the provability of the consistency of a mathematical theory and the decidability of statements by the axioms of the theory.
	at.yorku.ca /t/a/i/c/52.htm (1972 words)

	[No title]
		The negation of the fcp is the statement that a strong finite version of the compactness theorem holds.
		Woodin proved that $\Sigma^2_1$ statements are absolute for set forcing between models of "ZFC + CH + there are many large cardinals" (A $\Sigma^2_1$ statement is one which allows one existential quantification over sets of reals --- CH is a $\Sigma^2_1$ statement).
		Theorem (ZFC) Let G be a group and let F be the family of pairwise non-conjugate subgroups of G. If G is of uncountable cardinality $\lambda$ then the cardinality of F is at least $\lambda$.
	www.math.cmu.edu /~rami/seminar.past.html (4515 words)

	Kurt Gödel (Stanford Encyclopedia of Philosophy)
		This is because, as he points out, all the existential statements are based on his theorem V (giving the numeralwise expressibility of primitive recursive relations), which is intuitionistically unobjectionable.
		Another approach is to find a list of the main categories (e.g., causation, substance, action) and their interrelations, which, however, are to be arrived at phenomenologically.
		And in Gödel's list "My Notes, 1940–1970" he refers to the "main question" in philosophy as one which is bound up with the problem of evidence, by which problem he means, presumably, that of giving a precise characterization of it.
	plato.stanford.edu /entries/goedel (15214 words)

	Infinite Number [Definition] (Site not responding. Last check: 2007-11-06)
		If a particular theologian crosses over two centuries, they may be listed in the latter century or in the century in which they are best identified with.
		Certain extended number systems, such as the hyperreal numbersIn mathematics, particularly in non-standard analysis and mathematical logic, hyperreal numbers or nonstandard reals (usually denoted as *R) denote an ordered field which is a proper extension of the ordered field of real numbers R and which satisfies the transfer principle.
		A hyperbole, largely synonymous with exaggeration and overstatement, is a figure of speech in which statements are exaggerated or extravagant.
	www.wikimirror.com /Infinite_number (5324 words)

	Readings in Logic
		Since ZF (or ZFC) seems destined to stick around, a development of it in more-contemporary language is given by [Suppes1972].
		It is plain here and in (a) that explorations of mathematical logic moved into more-abstracted and intricate realms, seemingly far removed from the naive origins of axiomatic set theory.
		List of illustrations [p.188 and 218 entries are reversed]
	orcmid.com /readings/logic.htm (4265 words)

	Yuri Gurevich: Annotated Articles
		The theorems are used to explain and sharpen several recent undecidability results related to the corroboration problem, the simultaneous rigid E-unification problem and the prenex fragment of intuitionistic logic with equality.
		The observation that C expressions do not contain statements gives rise to the first evolving algebra which captures the command part of C; expressions are evaluated by an oracle.
		In the second chapter, the author completes the decision problem for (the prefix-vocabulary fragments of) pure logic of predicates and functions, though the treatment of the most difficult decidable class is deferred to 18.
	research.microsoft.com /~gurevich/annotated.html (14059 words)

	AMAST Links Vol. 02, Issue 01 - whole-issue
		Applications (including curriculum vitae, list of publications, research plan, names of references with their e-mail addresses, and intended period of stay) should be sent by the end of February 1995 to Kurt Mehlhorn or Michiel Smid.
		Abstracts of the papers listed below, together with more detailed information (reading advice, instructions for FTP retrieval of the full papers, etc.) are in the full version of this announcement.
		A list of the people presenting works and of their coauthors, in alphabetical order together with the list of their publications at these workshops.
	www.amast.org /archive/amast/links/v02/i01/AL0201.html (10089 words)

	Obsidian Wings: About Morality
		Besides, on most accounts of morality, a desire to prop up one's own conception of oneself by vilifying others is not high on the list of desirable motivations; and self-deception and hypocrisy are not among the characteristics we ought to tolerate, let alone cultivate.
		My understanding of it is that those statements are simply seen to be true through an understanding of their self-referentialness, not through the possibility of providing a formal derivation of them in some other system.
		It's potentially useful as tool in undecidability results -- for example, one of my friends showed that a particular lattice-theoretic question was undecidable by essentially encoding arithmetic into the lattices in question -- but there are oftentimes more tractable theories at hand.
	obsidianwings.blogs.com /obsidian_wings/2006/03/about_morality.html (13044 words)

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