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Topic: Axiom of infinity

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  PlanetMath: axiom schema of separation
The Axiom Schema of Separation is an axiom schema of Zermelo-Fraenkel set theory.
Another consequence of the Axiom Schema of Separation is that a subclass of any set is a set.
This is version 15 of axiom schema of separation, born on 2003-06-24, modified 2003-06-25.
planetmath.org /encyclopedia/Separation2.html   (184 words)

 Axiom of infinity - Wikipedia, the free encyclopedia
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory.
The axiom of infinity thus assumes the existence of this set.
The axiom of infinity is also one of the von Neumann-Bernays-Gödel axioms.
en.wikipedia.org /wiki/Axiom_of_infinity   (427 words)

 PlanetMath: axiom of infinity
The Axiom of Infinity is an axiom of Zermelo-Fraenkel set theory.
At first glance, this axiom seems to be ill-defined.
This is version 3 of axiom of infinity, born on 2003-07-03, modified 2003-07-06.
planetmath.org /encyclopedia/Infinity.html   (161 words)

 Axiom of empty set - Wikipedia, the free encyclopedia
In some formulations of ZF, the axiom of empty set is actually repeated in the axiom of infinity.
Also, the ZF axioms can also be written using a constant symbol representing the empty set; then the axiom of infinity uses this symbol without requiring it to be empty, while the axiom of empty set is needed to state that it is in fact empty.
That said, any axiom of set theory or logic that implies the existence of any set will imply the existence of the empty set, if one has the axiom schema of separation.
en.wikipedia.org /wiki/Axiom_of_empty_set   (322 words)

 Ernst Zermelo
His proof of the well-ordering theorem, which was based on the axiom of choice, was not accepted by all mathematicians, partly because the lack of axiomatization of set theory at this time.
In 1905, Zermelo began to axiomatise set theory and in 1908, he published his results despite his failure to prove consistency of his axiomatic system.
The resulting system, now called Zermelo-Fraenkel axioms (ZF), with ten axioms, is now the most commonly used one for axiomatic set theory.
www.ebroadcast.com.au /lookup/encyclopedia/ze/Zermelo.html   (439 words)

 Peano axioms Summary
In mathematics, the Peano axioms (or Peano postulates) are a set of second-order axioms proposed by Giuseppe Peano which determine the theory of arithmetic.
The axioms are usually encountered in a first-order form, where the crucial second-order induction axiom is replaced by an infinite first-order induction schema, and Peano Arithmetic (PA) is by convention the name of the widely used system of first-order arithmetic given using this first-order form.
The axiom of infinity guarantees the existence of an inductive set, so the set N is well-defined.
www.bookrags.com /Peano_axioms   (3389 words)

 Amazon.ca: Axiomatic Set Theory: Books: Patrick Suppes   (Site not responding. Last check: 2007-11-03)
The axiom schema that is used explicitly in the book is the "axiom schema of separation" due to Ernst Zermelo, which he formulated in order to make precise the notion of a statement as being "definite".
The axiom of infinity is brought in to permit the construction of arithmetical operations as certain sets.
The author shows that the use of this axiom allows one to prove that an infinite set has a denumerable subset, and he shows the equivalence of the axiom of choice with the numeration theorem, the well-ordering theorem, Zorn's lemma, and the law of trichotomy.
www.amazon.ca /Axiomatic-Set-Theory-Patrick-Suppes/dp/0486616304   (1706 words)

In addition to quantitative property and abstraction, a third important element is as equally important as the concept of infinity; infinity is based on the concept of possibility.
It is common to confuse infinity as quantity and produce language such as "infinitely large number", which is a mixture of possibleness and quantity.
The present explanation of infinity gives a nice way out for Russell's axiom of infinity, one of the two major problems of the logicists' program: "There are infinite number of things in the universe." This axiom was rejected because it is not logically sustainable.
www.usfca.edu /philosophy/pd1/discourse/8/bold4.html   (1120 words)

 MA10126 - Set Theory - Axioms of Set Theory (ZFC)
Axiom of Separation - If φ is a property with parameter p, then for any X and p there exists a set Y containing exactly those elements of X which satisfy property φ.
Axiom of the Power Set - For all sets X, there exists a set Y = P(X) whose elements are all the subsets of X. Georg Cantor used this axiom to prove that not all infinite sets have the same size (cardinality).
Stated like this, the Axiom of Choice appears intuitively obvious - but the axiom is equivalent to a number of statements that are strongly counter-intuitive - primarily the Well-Ordering Theorem, and the Banach-Tarski Paradox.
www.bath.ac.uk /~njs25/axioms.html   (607 words)

 mmtheorems46 - Metamath Proof Explorer
Axiom of Union, reproved from conditionless ZFC axioms.
Axiom of Regularity, reproved from conditionless ZFC axioms..
Axiom of Infinity, reproved from conditionless ZFC axioms.
metamath.planetmirror.com /mpegif/mmtheorems46.html   (690 words)

 [No title]
A strong axiom of infinity is asserted to enable conservative development of classical set theory and its metatheory, and other foundational work for which a strong metalanguage is convenient.
The axiom asserted talks as if the individuals were a kind of proto-set-theory, an alternative axiom mentioned but not asserted talks as if the individuals were proto-ordinals.
The infinity axiom does not distinguish any element to serve as an empty set or as the ordinal zero.
www.rbjones.com /rbjpub/pp/strong_infinity-m.html   (400 words)

The axiom of infinity, which states that the collection of all natural numbers forms a set, goes back only to the development of set theory which is much more recent.
Without the axiom of infinity, we can depend only on sets that can be developed from the empty set in a finite number of steps using these operations.
The links to previous theorems go to the proof of the theorem, but since the definitions and axioms do not have proofs, the links form them go to their statements in the main web page for the section in which they are found.
www.sonoma.edu /users/w/wilsonst/Papers/finite/forward.html   (714 words)

 Hector Parr's Essays: Infinity
The idea of infinity has been used increasingly over the centuries by mathematicians, and in general they have been more circumspect than have astronomers in their use of the word.
Cantor and others were not able to treat in a similar fashion their discussion of the equality or inequality of the different infinities they had discovered, but it must be conceded that their arguments had no relevence in the real world of material particles and real magnitudes.
Only when attempts are made to relate the concept of infinity to objects in the real world, as Russell did with his Axiom of Infinity, do we meet insurmountable logical contradictions.
www.c-parr.freeserve.co.uk /hcp/infinity.htm   (4801 words)

 Metamath Proof Explorer Home Page
When an axiom or theorem with a distinct variable condition is referenced in a proof, the distinct variable conditions attached the theorem being proved must satisfy those of the referenced axiom or theorem after substitutions are made into the referenced axiom or theorem.
The first three are the axiom and rule schemes for traditional predicate calculus, and the last two are the axiom schemes for the traditional theory of equality.
Although in some sense the traditional axiom schemes are more compact than Metamath's ax-4 through ax-16, their goal is simply to provide recipes for generating actual axioms, from which we then prove actual theorems.
de2.metamath.org /mpegif/mmset.html   (8321 words)

 What's an Axiom
In others the axioms cannot be sensibly denumerated (numerated in an ordinal sense with any sort of metric of ``closeness'') and are unique, disjoint, random.
If your philosophical axioms include a belief in God, and your memetic axioms include the particular interpretation of Leviticus that prohibits pastrami and provolone or bread made with milk in the same bite, well, the reuben is out.
If your personal axioms also include the laws of temporal continuity and causality (and hence, physics, biology, and all the rest), you might well conclude that hot shit on marble isn't likely to be either tasty or nutritious, leaving you with grilled cheese.
www.phy.duke.edu /~rgb/Philosophy/axioms/axioms/node3.html   (2002 words)

 Axiom of infinity
The integers are defined by an axiom that asserts the existence of a set
The remaining axioms are developed in the next chapter starting in Section 6.3.
The discussion of the infinite at the end of this chapter and the start of the next lays the groundwork for those axioms.
www.mtnmath.com /whatrh/node45.html   (50 words)

 Springer Online Reference Works
The existence of a natural numbers object, as a postulate, plays much the same role in topos theory as the axiom of infinity (see Infinity, axiom of) does in set theory.
In classical set theory, this axiom is normally viewed as giving rise to the incompleteness phenomenon (see Gödel incompleteness theorem), via Gödel numbering of formulas; however, in the constructive logic of toposes, the picture is rather different.
Using this, he was able to demonstrate the existence of a  "rewrite rule"  which converts any sentence in higher-order intuitionistic type theory with the axiom of infinity into a sentence in the corresponding theory without the axiom of infinity, in such a way that provability is preserved and reflected.
eom.springer.de /N/n120030.htm   (746 words)

 When axioms collide
The consequence is another 'euclidean' (as opposed to 'non-euclidean') geometry that uses another, and equally valid, axiom, permitting an infinite number of lines per planar point pair.
The ZF infinity axiom establishes a prototype infinite set, with the null set as its 'initial' element that permits a set
The fact that ZF permits the set J to exist and that it takes another axiom to knock it out means that J exists in another 'euclidean' geometry where the one-line-per-point-pair axiom is replaced.
www.geocities.com /n_fold/axiom.html   (481 words)

 The axioms of ZF
The axiom of replacement allows us to make a new set v from any statement in the language of ZF (with any fixed parameters) that defines y uniquely as a function of x and any set u.
This is the axiom that defines sets of higher cardinality or at least seems to.
The important thing to understand about the axioms is that they are comparatively simple precise rules for deducing new statements from existing ones.
www.mtnmath.com /book/node53.html   (456 words)

 Set Theory. Zermelo-Fraenkel Axioms. Russell's Paradox. Infinity. By K.Podnieks
The axioms C1, C1' and C2[F] (for all formulas F that do not contain x) and the axiom of choice define a formal set theory C which corresponds almost 100% to Cantor's intuitive set theory (of the "pre-paradox" period of 1873-94).
The axiom of infinity completes the list of comprehension axioms, which are necessary for reconstruction of common mathematics, i.e.
The set theory adopting the axiom of extensionality (C1), the axiom C1', the separation axiom schema (C21), the pairing axiom (C22), the union axiom (C23), the power-set axiom (C24), the replacement axiom schema (C25), the axiom of infinity (C26) and the axiom of regularity (C3), is called Zermelo-Fraenkel set theory, and is denoted by ZF.
linas.org /mirrors/www.ltn.lv/2005.01.29/~podnieks/gt2.html   (8496 words)

 Who's afraid of Kurt Gödel? | Ask MetaFilter   (Site not responding. Last check: 2007-11-03)
Not that it necessarily helps, but an alternative statement of the axiom of infinity is that "there exists a set that is a proper subset of its union" which at least dispenses with the natural numbers aspect.
As for math without the axiom of infinity, well, the vast majority of mathematicians think that all the axioms of ZFC are perfectly reasonable and very intuitive.
If you drop some of the axioms, then many of the things that mathematicians believe to be true are all of a sudden impossible to prove, and many other bits become much more messy.
ask.metafilter.com /mefi/29489   (2547 words)

 Infinity   (Site not responding. Last check: 2007-11-03)
One of the standard axioms of Zermelo-Frankel set theory, the axiom of infinity, asserts the existence of an infinite set.
This axiom is now accepted dogma but was highly controversial a hundred years ago.
Everyone now accepts the controversial axiom of infinity as a article of the mathematical faith.
www.math.hawaii.edu /~dale/godel/infinity.html   (552 words)

 SparkNotes: Bertrand Russell: Principia Mathematica
The first is the axiom of infinity, which postulates that there is an infinity of numbers.
Some critics have argued that the axiom of infinity is not a priori in nature but is an empirical question whose answer depends on experience.
This axiom is necessary to avoid Russell’s Paradox, but apart from that it does not seem to have a purely logical justification.
www.sparknotes.com /philosophy/russell/section1.html   (1726 words)

 Michael Potter - Set Theory and Its Philosophy: a Critical Introduction - Reviewed by Timothy Bays, University of Notre ...
Replacement is introduced as one of several axioms which govern the height of the set-theoretic hierarchy (the other main ones are the previously mentioned "axiom of ordinals" and an axiom scheme of reflection, although modern large cardinal axioms do receive a brief mention).
There are well-known difficulties with generating these axioms from the iterative conception (for instance, the very fact that the axioms don't follow from axiomatizations of the kind Potter gives in section one).
I think, for instance, that replacement looks more intuitive when it's considered in conjunction with the axiom of infinity, and that regressive arguments for choice and replacement together work better than arguments for either of them by itself (because, for instance, of the nice structure they jointly put on the classes of cardinals and ordinals).
ndpr.nd.edu /review.cfm?id=2141   (2338 words)

 Set Theory > Zermelo-Fraenkel Set Theory (Stanford Encyclopedia of Philosophy)
This axiom asserts that when sets x and y have the same members, they are the same set.
Since it is provable from this axiom and the previous axiom that there is a unique such set, we may introduce the notation ‘Ø’ to denote it.
Then the Axiom of Infinity asserts that there is a set x which contains Ø as a member and which is such that, anytime y is a member of x, then y∪{y} is a member of x.
plato.stanford.edu /entries/set-theory/ZF.html   (700 words)

 A/Prof N J Wildberger Personal Pages
Axiom of Foundation: Every nonempty set has a minimal element, that is one which does not contain another in the set.
Axiom of Choice: Every family of nonempty sets has a choice function, namely a function which assigns to each of the sets one of its elements.
With the notable exception of the `Axiom of Choice', I bet that fewer than 5% of mathematicians have ever employed even one of these `Axioms' explicitly in their published work.
web.maths.unsw.edu.au /~norman/views2.htm   (6444 words)

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