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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>. | ||
