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univalence_axiom [2016/07/18 00:27]
nikolaj
univalence_axiom [2016/07/18 01:02] (current)
nikolaj
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 In set theory, we can make //models// of other structures so that truths of their theories are reflected as truths of sets. For example, one commonly defines ${\bar 0}\equiv\{\}$,​ ${\bar S}x\equiv x\cup\{x\}$ and addition ${\bar +}$ in such a way that ${\bar S}{\bar S}{\bar 0}{\bar +}{\bar S}{\bar S}{\bar S}{\bar 0}=_{ZF}{\bar S}{\bar S}{\bar S}{\bar S}{\bar S}{\bar 0}$ can be proven. All those sets (representing numbers) are then collected and put into a set $\mathbb N_{ZF}\equiv\{{\bar 0},{\bar S}{\bar 0},{\bar S}{\bar S}{\bar 0},\dots\}$ In set theory, we can make //models// of other structures so that truths of their theories are reflected as truths of sets. For example, one commonly defines ${\bar 0}\equiv\{\}$,​ ${\bar S}x\equiv x\cup\{x\}$ and addition ${\bar +}$ in such a way that ${\bar S}{\bar S}{\bar 0}{\bar +}{\bar S}{\bar S}{\bar S}{\bar 0}=_{ZF}{\bar S}{\bar S}{\bar S}{\bar S}{\bar S}{\bar 0}$ can be proven. All those sets (representing numbers) are then collected and put into a set $\mathbb N_{ZF}\equiv\{{\bar 0},{\bar S}{\bar 0},{\bar S}{\bar S}{\bar 0},\dots\}$
  
-Another example is the model of the ordered pair $\langle x,​y\rangle\equiv \{\{x\},​\{x,​y\}\}$ and then the following desired statement can be proven: $(\langle x,​y\rangle=_{ZF}\langle u,​v\rangle)\Leftrightarrow (x=_{ZF}u\land y=_{ZF}v)$. Furthermore,​ a group can be modeled as a pair $\langle G,​*\rangle$,​ where is a set of group elements and $*$ is the function defining the multiplication table. Then the group $\mathbb Z_2$ of two elements can be presented as ${\mathrm z}_2^+\equiv\langle\{0,​1\},​+\rangle$ or ${\mathrm z}_2^\times\equiv\langle\{1,​-1\},​\times\rangle$,​ where $+$ is arithmetic addition mod $2$ and $\times$ is arithmetic multiplication.+Another example is the model of the ordered pair $\langle x,​y\rangle\equiv \{\{x\},​\{x,​y\}\}$ and then the following desired statement can be proven: $(\langle x,​y\rangle=_{ZF}\langle u,​v\rangle)\Leftrightarrow (x=_{ZF}u\land y=_{ZF}v)$. Furthermore,​ a group can be modeled as a pair $\langle G,​*\rangle$,​ where $G$ is a set of group elements and $*$ is the function defining the multiplication table. Then the group $\mathbb Z_2$ of two elements can be presented as ${\mathrm z}_2^+\equiv\langle\{0,​1\},​+\rangle$ or ${\mathrm z}_2^\times\equiv\langle\{1,​-1\},​\times\rangle$,​ where $+$ is arithmetic addition mod $2$ and $\times$ is arithmetic multiplication.
  
 Because of its capacity to define models set theory can be taken as //​foundation for mathematics//​. One only needs <logic and set theory axioms and definition of concepts in terms of sets> as opposed to <logic and new axioms for each theory>​. ​ Because of its capacity to define models set theory can be taken as //​foundation for mathematics//​. One only needs <logic and set theory axioms and definition of concepts in terms of sets> as opposed to <logic and new axioms for each theory>​. ​
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