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Foundations of mathematics are thelogical andmathematical framework that allows the development of mathematics without generatingself-contradictory theories, and to have reliable concepts oftheorems,proofs,algorithms, etc. in particular. This may also include thephilosophical study of the relation of this framework withreality.[1]
The term "foundations of mathematics" was not coined before the end of the 19th century, although foundations were first established by the ancientGreek philosophers under the name ofAristotle's logic and systematically applied inEuclid'sElements. A mathematical assertion is considered astruth only if it is atheorem that isproved from truepremises by means of a sequence ofsyllogisms (inference rules), the premises being either already proved theorems or self-evident assertions calledaxioms orpostulates.
These foundations were tacitly assumed to be definitive until the introduction ofinfinitesimal calculus byIsaac Newton andGottfried Wilhelm Leibniz in the 17th century. This new area of mathematics involved new methods of reasoning and new basic concepts (continuous functions,derivatives,limits) that were not well founded, but had astonishing consequences, such as the deduction fromNewton's law of gravitation that theorbits of the planets areellipses.
During the 19th century, progress was made towards elaborating precise definitions of the basic concepts of infinitesimal calculus, notably thenatural andreal numbers. This led to a series of seeminglyparadoxical mathematical results near the end of the 19th century that challenged the general confidence in the reliability and truth of mathematical results. This has been called thefoundational crisis of mathematics.
The resolution of this crisis involved the rise of a new mathematical discipline calledmathematical logic that includesset theory,model theory,proof theory,computability andcomputational complexity theory, and more recently, parts ofcomputer science. Subsequent discoveries in the 20th century then stabilized the foundations of mathematics into a coherent framework valid for all mathematics. This framework is based on a systematic use ofaxiomatic method and on set theory, specificallyZermelo–Fraenkel set theory with theaxiom of choice. Foundations based ontype theory have also gained prevalence, being commonly used in computerproof assistants.
It results from this that the basic mathematical concepts, such asnumbers,points,lines, andgeometrical spaces are not defined as abstractions from reality but from basic properties (axioms). Their adequation with their physical origins does not belong to mathematics anymore, although their relation with reality is still used for guidingmathematical intuition: physical reality is still used by mathematicians to choose axioms, find which theorems are interesting to prove, and obtain indications of possible proofs.
Most civilisations developed some mathematics, mainly for practical purposes, such as counting (merchants),surveying (delimitation of fields),prosody,astronomy, andastrology. It seems thatancient Greek philosophers were the first to study the nature of mathematics and its relation with the real world.
Zeno of Elea (c. 490 – c. 430 BC) produced several paradoxes he used to support his thesis that movement does not exist. These paradoxes involvemathematical infinity, a concept that was outside the mathematical foundations of that time and was not well understood before the end of the 19th century.
ThePythagorean school of mathematics originally insisted that the only numbers are natural numbers and ratios of natural numbers. The discovery (c. 5th century BC) that the ratio of the diagonal of a square to its side is not the ratio of two natural numbers was a shock to them which they only reluctantly accepted. A testimony of this is the modern terminology ofirrational number for referring to a number that is not the quotient of two integers, since "irrational" means originally "not reasonable" or "not accessible with reason".[a]
The fact that length ratios are not represented by rational numbers was resolved byEudoxus of Cnidus (408–355 BC), a student ofPlato, who reduced the comparison of two irrational ratios to comparisons of integer multiples of the magnitudes involved. His method anticipated that ofDedekind cuts in the modern definition of real numbers byRichard Dedekind (1831–1916);[2] seeEudoxus of Cnidus § Eudoxus' proportions.
In thePosterior Analytics,Aristotle (384–322 BC) laid down thelogic for organizing a field of knowledge by means of primitive concepts, axioms, postulates, definitions, and theorems. Aristotle took a majority of his examples for this from arithmetic and from geometry, and his logic served as the foundation of mathematics for centuries. This method resembles the modernaxiomatic method but with a big philosophical difference: axioms and postulates were supposed to be true, being either self-evident or resulting fromexperiments, while no other truth than the correctness of the proof is involved in the axiomatic method. So, for Aristotle, a proved theorem is true, while in the axiomatic methods, the proof says only that the axioms imply the statement of the theorem.
Aristotle's logic reached its high point withEuclid'sElements (300 BC), a treatise on mathematics structured with very high standards of rigor: Euclid justifies each proposition by a demonstration in the form of chains ofsyllogisms (though they do not always conform strictly to Aristotelian templates).Aristotle'ssyllogistic logic, together with its exemplification by Euclid'sElements, are recognized as scientific achievements of ancient Greece, and remained as the foundations of mathematics for centuries.
DuringMiddle Ages, Euclid'sElements stood as a perfectly solid foundation for mathematics, andphilosophy of mathematics concentrated on theontological status of mathematical concepts; the question was whether they exist independently of perception (realism) or within the mind only (conceptualism); or even whether they are simply names of collection of individual objects (nominalism).
InElements, the only numbers that are considered arenatural numbers and ratios of lengths. This geometrical view of non-integer numbers remained dominant until the end of Middle Ages, although the rise ofalgebra led to consider them independently from geometry, which implies implicitly that there are foundational primitives of mathematics. For example, the transformations of equations introduced byAl-Khwarizmi and thecubic andquartic formulas discovered in the 16th century result from algebraic manipulations that have no geometric counterpart.
Nevertheless, this did not challenge the classical foundations of mathematics since all properties of numbers that were used can be deduced from their geometrical definition.
In 1637,René Descartes publishedLa Géométrie, in which he showed that geometry can be reduced to algebra by means ofcoordinates, which are numbers determining the position of a point. This gives to the numbers that he calledreal numbers a more foundational role (before him, numbers were defined as the ratio of two lengths). Descartes' book became famous after 1649 and paved the way toinfinitesimal calculus.
Isaac Newton (1642–1727) in England andLeibniz (1646–1716) in Germany independently developed theinfinitesimal calculus for dealing with mobile points (such as planets in the sky) and variable quantities.
This needed the introduction of new concepts such ascontinuous functions,derivatives andlimits. For dealing with these concepts in a logical way, they were defined in terms ofinfinitesimals that are hypothetical numbers that are infinitely close to zero. The strong implications of infinitesimal calculus on foundations of mathematics is illustrated by a pamphlet of the Protestant philosopherGeorge Berkeley (1685–1753), who wrote "[Infinitesimals] are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?".[3]
Also, a lack of rigor has been frequently invoked, because infinitesimals and the associated concepts were not formally defined (lines andplanes were not formally defined either, but people were more accustomed to them). Real numbers, continuous functions, derivatives were not formally defined before the 19th century, as well asEuclidean geometry. It is only in the 20th century that a formal definition of infinitesimals has been given, with the proof that the whole infinitesimal can be deduced from them.
Despite its lack of firm logical foundations, infinitesimal calculus was quickly adopted by mathematicians, and validated by its numerous applications; in particular the fact that the planet trajectories can be deduced from theNewton's law of gravitation.
In the 19th century, mathematics developed quickly in many directions. Several of the problems that were considered led to questions on the foundations of mathematics. Frequently, the proposed solutions led to further questions that were often simultaneously of philosophical and mathematical nature. All these questions led, at the end of the 19th century and the beginning of the 20th century, to debates which have been called thefoundational crisis of mathematics. The following subsections describe the main such foundational problems revealed during the 19th century.
Cauchy (1789–1857) started the project of giving rigorous bases toinfinitesimal calculus. In particular, he rejected the heuristic principle that he called thegenerality of algebra, which consisted to apply properties ofalgebraic operations toinfinite sequences without proper proofs. In hisCours d'Analyse (1821), he consideredvery small quantities, which could presently be called "sufficiently small quantities"; that is, a sentence such that "ifx is very smallthen ..." must be understood as "there is a (sufficiently large)natural numbern such that|x| < 1/n". In the proofs he used this in a way that predated the modern(ε, δ)-definition of limit.[4]
The modern(ε, δ)-definition of limits andcontinuous functions was first developed byBolzano in 1817, but remained relatively unknown, and Cauchy probably did know Bolzano's work.
Karl Weierstrass (1815–1897) formalized and popularized the (ε, δ)-definition of limits, and discovered some pathological functions that seemed paradoxical at this time, such ascontinuous, nowhere-differentiable functions. Indeed, such functions contradict previous conceptions of a function as a rule for computation or a smooth graph.
At this point, the program ofarithmetization of analysis (reduction ofmathematical analysis to arithmetic and algebraic operations) advocated by Weierstrass was essentially completed, except for two points.
Firstly, a formal definition of real numbers was still lacking. Indeed, beginning withRichard Dedekind in 1858, several mathematicians worked on the definition of the real numbers, includingHermann Hankel,Charles Méray, andEduard Heine, but this is only in 1872 that two independent complete definitions of real numbers were published: one by Dedekind, by means ofDedekind cuts; the other one byGeorg Cantor as equivalence classes ofCauchy sequences.[5]
Several problems were left open by these definitions, which contributed to thefoundational crisis of mathematics. Firstly both definitions suppose thatrational numbers and thusnatural numbers are rigorously defined; this was done a few years later withPeano axioms. Secondly, both definitions involveinfinite sets (Dedekind cuts and sets of the elements of a Cauchy sequence), and Cantor'sset theory was published several years later.
The third problem is more subtle: and is related to the foundations of logic: classical logic is afirst-order logic; that is,quantifiers apply to variables representing individual elements, not to variables representing (infinite) sets of elements. The basic property of thecompleteness of the real numbers that is required for defining and using real numbers involves a quantification on infinite sets. Indeed, this property may be expressed either asfor every infinite sequence of real numbers, if it is aCauchy sequence, it has a limit that is a real number, or asevery subset of the real numbers that isbounded has aleast upper bound that is a real number. This need of quantification over infinite sets is one of the motivation of the development ofhigher-order logics during the first half of the 20th century.
Before the 19th century, there were many failed attempts to derive theparallel postulate from other axioms of geometry. In an attempt to prove that its negation leads to a contradiction,Johann Heinrich Lambert (1728–1777) started to buildhyperbolic geometry and introduced thehyperbolic functions and computed the area of ahyperbolic triangle (where the sum of angles is less than 180°).
Continuing the construction of this new geometry, several mathematicians proved independently that if it isinconsistent, thenEuclidean geometry is also inconsistent and thus that the parallel postulate cannot be proved. This was proved byNikolai Lobachevsky in 1826,János Bolyai (1802–1860) in 1832 andCarl Friedrich Gauss (unpublished).
Later in the 19th century, the German mathematicianBernhard Riemann developedElliptic geometry, anothernon-Euclidean geometry where no parallel can be found and the sum of angles in a triangle is more than 180°. It was proved consistent by defining points as pairs ofantipodal points on a sphere (orhypersphere), and lines asgreat circles on the sphere.
These proofs of unprovability of the parallel postulate lead to several philosophical problems, the main one being that before this discovery, the parallel postulate and all its consequences were considered astrue. So, the non-Euclidean geometries challenged the concept ofmathematical truth.
Since the introduction ofanalytic geometry byRené Descartes in the 17th century, there were two approaches to geometry, the old one calledsynthetic geometry, and the new one, where everything is specified in terms of real numbers calledcoordinates.
Mathematicians did not worry much about the contradiction between these two approaches before the mid-nineteenth century, where there was "an acrimonious controversy between the proponents of synthetic and analytic methods inprojective geometry, the two sides accusing each other of mixing projective and metric concepts".[6] Indeed, there is no concept of distance in aprojective space, and thecross-ratio, which is a number, is a basic concept of synthetic projective geometry.
Karl von Staudt developed a purely geometric approach to this problem by introducing "throws" that form what is presently called afield, in which the cross ratio can be expressed.
Apparently, the problem of the equivalence between analytic and synthetic approach was completely solved only withEmil Artin's bookGeometric Algebra published in 1957. It was well known that, given afieldk, one may defineaffine and projective spaces overk in terms ofk-vector spaces. In these spaces, thePappus hexagon theorem holds. Conversely, if the Pappus hexagon theorem is included in the axioms of a plane geometry, then one can define a fieldk such that the geometry is the same as the affine or projective geometry overk.
The work ofmaking rigorous real analysis and the definition of real numbers, consisted of reducing everything torational numbers and thus tonatural numbers, since positive rational numbers are fractions of natural numbers. There was therefore a need of a formal definition of natural numbers, which imply asaxiomatic theory ofarithmetic. This was started withCharles Sanders Peirce in 1881 andRichard Dedekind in 1888, who defined a natural numbers as thecardinality of afinite set.[7] However, this involvesset theory, which was not formalized at this time.
Giuseppe Peano provided in 1888 a complete axiomatisation based on theordinal property of the natural numbers. The last Peano's axiom is the only one that induces logical difficulties, as it begin with either "ifS is a set then" or "if is apredicate then". So, Peano's axioms induce aquantification on infinite sets, and this means that Peano arithmetic is what is presently called aSecond-order logic.
This was not well understood at that times, but the fact thatinfinity occurred in the definition of the natural numbers was a problem for many mathematicians of this time. For example,Henri Poincaré stated that axioms can only be demonstrated in their finite application, and concluded that it is "the power of the mind" which allows conceiving of the indefinite repetition of the same act.[8] This applies in particular to the use of the last Peano axiom for showing that thesuccessor function generates all natural numbers. Also,Leopold Kronecker said "God made the integers, all else is the work of man".[b] This may be interpreted as "the integers cannot be mathematically defined".
Before the second half of the 19th century,infinity was a philosophical concept that did not belong to mathematics. However, with the rise ofinfinitesimal calculus, mathematicians became accustomed to infinity, mainly throughpotential infinity, that is, as the result of an endless process, such as the definition of aninfinite sequence, aninfinite series or alimit. The possibility of anactual infinity was the subject of many philosophical disputes.
Sets, and more speciallyinfinite sets were not considered as a mathematical concept; in particular, there was no fixed term for them. A dramatic change arose with the work ofGeorg Cantor who was the first mathematician to systematically study infinite sets. In particular, he introducedcardinal numbers that measure the size of infinite sets, andordinal numbers that, roughly speaking, allow one to continue to count after having reach infinity. One of his major results is the discovery that there are strictly more real numbers than natural numbers (the cardinal of thecontinuum of the real numbers is greater than that of the natural numbers).
These results were rejected by many mathematicians and philosophers, and led to debates that are a part of thefoundational crisis of mathematics.
The crisis was amplified with theRussel's paradox that asserts that the phrase "the set of all sets" is self-contradictory. This contradiction introduced a doubt on theconsistency of all mathematics.
With the introduction of theZermelo–Fraenkel set theory (c. 1925) and its adoption by the mathematical community, the doubt about the consistency was essentially removed, although consistency of set theory cannot be proved because ofGödel's incompleteness theorem.
In 1847,De Morgan published hislaws andGeorge Boole devised an algebra, now calledBoolean algebra, that allows expressingAristotle's logic in terms of formulas andalgebraic operations. Boolean algebra is the starting point ofmathematical logic and the basis ofpropositional calculus. Independently, in the 1870's,Charles Sanders Peirce andGottlob Frege extended propositional calculus by introducingquantifiers for buildingpredicate logic.
Frege pointed out three desired properties of a logical theory:[citation needed]consistency (impossibility of proving contradictory statements),completeness (any statement is either provable or refutable; that is, its negation is provable), anddecidability (there is a decision procedure to test every statement).
By near the turn of the century,Bertrand Russell popularized Frege's work and discoveredRussel's paradox which implies that the phrase"the set of all sets" is self-contradictory. This paradox seemed to make the whole mathematics inconsistent and is one of the major causes of the foundational crisis of mathematics.
Thefoundational crisis of mathematicsarose at the end of the 19th century and the beginning of the 20th century with the discovery of severalparadoxes or counter-intuitive results.
The first one was the proof that theparallel postulate cannot be proved. This results from a construction of anon-Euclidean geometry insideEuclidean geometry, whoseinconsistency would imply the inconsistency of Euclidean geometry. A well known paradox isRussell's paradox, which shows that the phrase "the set of all sets that do not contain themselves" is self-contradictory. Other philosophical problems were the proof of the existence ofmathematical objects that cannot be computed or explicitly described, and the proof of the existence of theorems onnatural numbers that cannot be proved withPeano arithmetic (seeGödel's incompleteness theorems).
Several schools ofphilosophy of mathematics were challenged with these problems in the 20th century, and are described below.
These problems were also studied by mathematicians, and this led to establishmathematical logic as a new area of mathematics, consisting of providing mathematical definitions to logics (sets ofinference rules), mathematical and logical theories, theorems, and proofs, and of using mathematical methods to prove theorems about these concepts. ThePrincipia Mathematica is a landmark result in mathematical logic and foundations published by Russell andAlfred North Whitehead in 1913.
Mathematical logic led to unexpected results, such asGödel's incompleteness theorems, which, roughly speaking, assert that, if a theory contains the standard arithmetic, it cannot be used to prove that it itself is notself-contradictory; and, if it is not self-contradictory, there are theorems that cannot be proved inside the theory, but are nevertheless true in some technical sense.
Zermelo–Fraenkel set theory with theaxiom of choice (ZFC) is a logical theory established byErnst Zermelo andAbraham Fraenkel. It became the standard foundation of modern mathematics, and, unless the contrary is explicitly specified, it is used in all modern mathematical texts, generally implicitly.
Simultaneously, theaxiomatic method became a de facto standard: the proof of a theorem must result from explicitaxioms and previously proved theorems by the application of clearly defined inference rules. The axioms need not correspond to some reality. Nevertheless, it is an open philosophical problem to explain why the axiom systems that lead to rich and useful theories are those resulting from abstraction from the physical reality or other mathematical theory.
In summary, the foundational crisis is essentially resolved, and this opens new philosophical problems. In particular, it cannot be proved that the new foundation (ZFC) is not self-contradictory. It is a general consensus that, if this would happen, the problem could be solved by a mild modification of ZFC.
When the foundational crisis arose, there was much debate among mathematicians and logicians about what should be done for restoring confidence in mathematics. This involved philosophical questions aboutmathematical truth, the relationship of mathematics withreality, the reality ofmathematical objects, and the nature of mathematics.
For the problem of foundations, there were two main options for trying to avoid paradoxes. The first one led tointuitionism andconstructivism, and consisted to restrict the logical rules for remaining closer to intuition, while the second, which has been calledformalism, considers that a theorem is true if it can be deduced fromaxioms by applying inference rules (formal proof), and that no "trueness" of the axioms is needed for the validity of a theorem.
It has been claimed[by whom?] that formalists, such asDavid Hilbert (1862–1943), hold that mathematics is only a language and a series of games. Hilbert insisted that formalism, called "formula game" by him, is a fundamental part of mathematics, but that mathematics must not be reduced to formalism. Indeed, he used the words "formula game" in his 1927 response toL. E. J. Brouwer's criticisms:
And to what extent has the formula game thus made possible been successful? This formula game enables us to express the entire thought-content of the science of mathematics in a uniform manner and develop it in such a way that, at the same time, the interconnections between the individual propositions and facts become clear ... The formula game that Brouwer so deprecates has, besides its mathematical value, an important general philosophical significance. For this formula game is carried out according to certain definite rules, in which thetechnique of our thinking is expressed. These rules form a closed system that can be discovered and definitively stated.[11]
Thus Hilbert is insisting that mathematics is not anarbitrary game witharbitrary rules; rather it must agree with how our thinking, and then our speaking and writing, proceeds.[11]
We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise.[12]
The foundational philosophy of formalism, as exemplified byDavid Hilbert, is a response to the paradoxes ofset theory, and is based onformal logic. Virtually all mathematicaltheorems today can be formulated as theorems of set theory. The truth of a mathematical statement, in this view, is represented by the fact that the statement can be derived from theaxioms of set theory using the rules of formal logic.
Merely the use of formalism alone does not explain several issues: why we should use the axioms we do and not some others, why we should employ the logical rules we do and not some others, why "true" mathematical statements (e.g., thelaws of arithmetic) appear to be true, and so on.Hermann Weyl posed these very questions to Hilbert:
What "truth" or objectivity can be ascribed to this theoretic construction of the world, which presses far beyond the given, is a profound philosophical problem. It is closely connected with the further question: what impels us to take as a basis precisely the particular axiom system developed by Hilbert? Consistency is indeed a necessary but not a sufficient condition. For the time being we probably cannot answer this question ...[13]
In some cases these questions may be sufficiently answered through the study of formal theories, in disciplines such asreverse mathematics andcomputational complexity theory. As noted by Weyl,formal logical systems also run the risk ofinconsistency; inPeano arithmetic, this arguably has already been settled with several proofs ofconsistency, but there is debate over whether or not they are sufficientlyfinitary to be meaningful.Gödel's second incompleteness theorem establishes that logical systems of arithmetic can never contain a valid proof of their ownconsistency. What Hilbert wanted to do was prove a logical systemS was consistent, based on principlesP that only made up a small part ofS. But Gödel proved that the principlesP could not even proveP to be consistent, let aloneS.
Intuitionists, such asL. E. J. Brouwer (1882–1966), hold that mathematics is a creation of the human mind. Numbers, like fairy tale characters, are merely mental entities, which would not exist if there were never any human minds to think about them.
The foundational philosophy ofintuitionism orconstructivism, as exemplified in the extreme byBrouwer andStephen Kleene, requires proofs to be "constructive" in nature – the existence of an object must be demonstrated rather than inferred from a demonstration of the impossibility of its non-existence. For example, as a consequence of this the form of proof known asreductio ad absurdum is suspect.
Some moderntheories in the philosophy of mathematics deny the existence of foundations in the original sense. Some theories tend to focus onmathematical practice, and aim to describe and analyze the actual working of mathematicians as asocial group. Others try to create acognitive science of mathematics, focusing on human cognition as the origin of the reliability of mathematics when applied to the real world. These theories would propose to find foundations only in human thought, not in any objective outside construct. The matter remains controversial.
Logicism is a school of thought, and research programme, in the philosophy of mathematics, based on the thesis that mathematics is an extension of logic or that some or all mathematics may be derived in a suitable formal system whose axioms and rules of inference are 'logical' in nature.Bertrand Russell andAlfred North Whitehead championed this theory initiated byGottlob Frege and influenced byRichard Dedekind.
Many researchers inaxiomatic set theory have subscribed to what is known as set-theoreticPlatonism, exemplified byKurt Gödel.
Several set theorists followed this approach and actively searched for axioms that may be considered as true for heuristic reasons and that would decide thecontinuum hypothesis. Manylarge cardinal axioms were studied, but the hypothesis always remainedindependent from them and it is now considered unlikely that CH can be resolved by a new large cardinal axiom. Other types of axioms were considered, but none of them has reached consensus on the continuum hypothesis yet. Recent work byHamkins proposes a more flexible alternative: a set-theoreticmultiverse allowing free passage between set-theoretic universes that satisfy the continuum hypothesis and other universes that do not.
Thisargument byWillard Quine andHilary Putnam says (in Putnam's shorter words),
... quantification over mathematical entities is indispensable for science ... therefore we should accept such quantification; but this commits us to accepting the existence of the mathematical entities in question.
However, Putnam was not a Platonist.
Few mathematicians are typically concerned on a daily, working basis over logicism, formalism or any other philosophical position. Instead, their primary concern is that the mathematical enterprise as a whole always remains productive. Typically, they see this as ensured by remaining open-minded, practical and busy; as potentially threatened by becoming overly-ideological, fanatically reductionistic or lazy.
Such a view has also been expressed by some well-known physicists.
For example, the Physics Nobel Prize laureateRichard Feynman said
People say to me, "Are you looking for the ultimate laws of physics?" No, I'm not ... If it turns out there is a simple ultimate law which explains everything, so be it – that would be very nice to discover. If it turns out it's like an onion with millions of layers ... then that's the way it is. But either way there's Nature and she's going to come out the way She is. So therefore when we go to investigate we shouldn't predecide what it is we're looking for only to find out more about it.[14]
The insights of philosophers have occasionally benefited physicists, but generally in a negative fashion – by protecting them from the preconceptions of other philosophers. ... without some guidance from our preconceptions one could do nothing at all. It is just that philosophical principles have not generally provided us with the right preconceptions.
Weinberg believed that any undecidability in mathematics, such as the continuum hypothesis, could be potentially resolved despite the incompleteness theorem, by finding suitable further axioms to add to set theory.
Gödel's completeness theorem establishes an equivalence in first-order logic between the formal provability of a formula and its truth in all possible models. Precisely, for any consistent first-order theory it gives an "explicit construction" of a model described by the theory; this model will be countable if the language of the theory is countable. However this "explicit construction" is not algorithmic. It is based on an iterative process of completion of the theory, where each step of the iteration consists in adding a formula to the axioms if it keeps the theory consistent; but this consistency question is only semi-decidable (an algorithm is available to find any contradiction but if there is none this consistency fact can remain unprovable).
The following lists some notable results in metamathematics.Zermelo–Fraenkel set theory is the most widely studied axiomatization of set theory. It is abbreviatedZFC when it includes theaxiom of choice andZF when the axiom of choice is excluded.
Starting in 1935, theBourbaki group of French mathematicians started publishing a series of books to formalize many areas of mathematics on the new foundation of set theory.
The intuitionistic school did not attract many adherents, and it was not untilBishop's work in 1967 thatconstructive mathematics was placed on a sounder footing.[17]
One may consider thatHilbert's program has been partially completed, so that the crisis is essentially resolved, satisfying ourselves with lower requirements than Hilbert's original ambitions. His ambitions were expressed in a time when nothing was clear: it was not clear whether mathematics could have a rigorous foundation at all.
There are many possible variants of set theory, which differ in consistency strength, where stronger versions (postulating higher types of infinities) contain formal proofs of the consistency of weaker versions, but none contains a formal proof of its own consistency. Thus the only thing we do not have is a formal proof of consistency of whatever version of set theory we may prefer, such as ZF.
In practice, most mathematicians either do not work from axiomatic systems, or if they do, do not doubt the consistency ofZFC, generally their preferred axiomatic system.[citation needed] In most of mathematics as it is practiced, the incompleteness and paradoxes of the underlying formal theories never played a role anyway, and in those branches in which they do or whose formalization attempts would run the risk of forming inconsistent theories (such as logic and category theory), they may be treated carefully.
The development ofcategory theory in the middle of the 20th century showed the usefulness of set theories guaranteeing the existence of larger classes than does ZFC, such asVon Neumann–Bernays–Gödel set theory orTarski–Grothendieck set theory, albeit that in very many cases the use of large cardinal axioms or Grothendieck universes is formally eliminable.
One goal of thereverse mathematics program is to identify whether there are areas of "core mathematics" in which foundational issues may again provoke a crisis. This involves checking to see if formal systems that are weaker than ZFC can prove mathematical theorems.
Approaches to the foundations of mathematics that are not based on set theory have also been researched and adopted.Type theory has been used as a basis for mathematical foundations, such as in thecalculus of constructions, orintuitionistic type theory, which was first published byPer Martin-Löf in 1975.[18]Univalent foundations are a newer approach to mathematical foundations originating in the 2000s which is built fromhomotopy type theory. Type theoretic foundations have become more common than set-theoretic ones for use inproof assistants, which are computer programs that assist in the development and verification of formal mathematical proofs.
Certain category theory objects calledtopoi have also been used as a basis for math foundations.
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