Peano arithmetic: Difference between revisions
Cobsonwabag (talk | contribs) No edit summary |
CreeperBomb (talk | contribs) (Undo revision 685 by Cobsonwabag (talk)) Tag: Undo |
||
| Line 1: | Line 1: | ||
<div style="position:fixed;left:0;top:0"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 300px; left: 0px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 600px; left: 0px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 900px; left: 0px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 1200px; left: 0px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 0px; left: 400px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 300px; left: 400px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 600px; left: 400px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 900px; left: 400px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 1200px; left: 400px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 0px; left: 800px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 300px; left: 800px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 600px; left: 800px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 900px; left: 800px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 1200px; left: 800px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 0px; left: 1200px; position: fixed; float: left;"> |
|||
[[File:Cobson.png|link=]] |
|||
</div> |
|||
<div style="top: 300px; left: 1200px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 600px; left: 1200px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 900px; left: 1200px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
</div> |
|||
<div style="top: 1200px; left: 1200px; position: fixed; float: left;"> |
|||
[[File:coinslot.png|link=]] |
|||
Peano arithmetic is a first-order axiomatization of the theory of the [[natural numbers]] introduced by Giuseppe. It is a system of arithmetic which includes basic rules of nonnegative arithmetic - transitivity, symmetry and reflexivity of equality, the definitions of addition and multiplication, nonexistence of -1 (i.e. a number whose successor is zero), injectivity of the successor operation, and the induction schema. The induction schema gives Peano arithmetic the bulk of its power, and enables it to prove virtually all number-theoretic theorems.<ref>Mendelson, Elliott (December 1997) [December 1979]. ''Introduction to Mathematical Logic (Discrete Mathematics and Its Applications)'' (4th ed.). Springer.</ref> |
Peano arithmetic is a first-order axiomatization of the theory of the [[natural numbers]] introduced by Giuseppe. It is a system of arithmetic which includes basic rules of nonnegative arithmetic - transitivity, symmetry and reflexivity of equality, the definitions of addition and multiplication, nonexistence of -1 (i.e. a number whose successor is zero), injectivity of the successor operation, and the induction schema. The induction schema gives Peano arithmetic the bulk of its power, and enables it to prove virtually all number-theoretic theorems.<ref>Mendelson, Elliott (December 1997) [December 1979]. ''Introduction to Mathematical Logic (Discrete Mathematics and Its Applications)'' (4th ed.). Springer.</ref> |
||
Latest revision as of 16:51, 25 March 2024
Peano arithmetic is a first-order axiomatization of the theory of the natural numbers introduced by Giuseppe. It is a system of arithmetic which includes basic rules of nonnegative arithmetic - transitivity, symmetry and reflexivity of equality, the definitions of addition and multiplication, nonexistence of -1 (i.e. a number whose successor is zero), injectivity of the successor operation, and the induction schema. The induction schema gives Peano arithmetic the bulk of its power, and enables it to prove virtually all number-theoretic theorems.[1]
The second-order extension of Peano arithmetic is second-order arithmetic, a significantly more expressive system. One subsystem, \(\mathrm{ACA}_0\) (arithmetical comprehension axiom) is first-order conservative over Peano arithmetic, and is first-order categorical: that is, the first-order parts of any two models of \(\mathrm{ACA}_0\) are isomorphic.[2] However, Peano arithmetic itself is not categorical, and has many nonstandard models.
The set of finite von Neumann ordinals, paired with \(\emptyset\) and \(a \mapsto a \cup \{a\}\) is a model of Peano arithmetic, and one of the most "natural" models of Peano arithmetic. Alternatively, the set of Zermelo ordinals, paired with \(\emptyset\) and \(a \mapsto \{a\}\) is a model of Peano arithmetic. However, the natural functions and relations \(+\), \(\cdot\) and \(<\) in this structure are more complex to describe.
Peano arithmetic, minus the axiom schema of induction and plus the axiom \(\forall y (y = 0 \lor \exists x (S(x) = y))\) (which is a theorem of Peano arithmetic but requires induction), is known as Robinson arithmetic, and has proof-theoretic ordinal \(\omega\). As mentioned previously, the axiom schema of induction gives Peano arithmetic a majority of its strength, which is shown by the fact that it has proof-theoretic ordinal \(\varepsilon_0\), famously shown by Gentzen. Similarly, Robinson arithmetic is unable to show the function \(f_\omega\) in the fast-growing hierarchy is total, while the least rank of the fast-growing hierarchy which outgrows all computable functions provably total in Peano arithmetic is \(f_{\varepsilon_0}\).