Axiom

An ax­iom of a the­ory \(T\) is a well-formed sen­tence in the lan­guage of the the­ory that we as­sume to be true with­out a for­mal jus­tifi­ca­tion.

Models

Models of a cer­tain the­ory are go­ing to be those math­e­mat­i­cal ob­jects in which the ax­ioms hold, so they can be used to pin down the math­e­mat­i­cal struc­tures we want to talk about.

Nor­mally, when we want to rea­son about a par­tic­u­lar as­pect of the world we have to try to figure out a suffi­ciently de­scrip­tive set of ax­ioms which are satis­fied by the thing we want to rea­son about. Then we can use de­duc­tion rules to de­duce con­se­quences of those ax­ioms, which will also be satis­fied by the thing in ques­tion.

For ex­am­ple, we may want to model how viral videos spread across the in­ter­net. Then we can make some as­sump­tions about this situ­a­tion. For ex­am­ple, we may con­sider that the in­ter­net is a graph in which each node is a per­son, and its edges are friend­ships. We may fur­ther as­sume that the edges have a weight be­tween 0 and 1 rep­re­sent­ing the prob­a­bil­ity that in a time step a per­son will tell its friend about the video. Then we can use this model to figure out how kit­ten videos end up on your twit­ter feed.

This is a par­tic­u­larly com­plex model with many as­sump­tions be­hind. For­mal­iz­ing all those as­sump­tions and turn­ing them into ax­ioms would be a pain in the ass, but they are still there, albeit hid­den.

For ex­am­ple, there might be an ax­iom in the lan­guage of first or­der logic stat­ing that \(\forall w. weight(w)\rightarrow 0; that is, every weight in the graph is between \(0\) and \(1\).

In the ideal case, we want to write down enough ax­ioms so that the only model satis­fy­ing them is the one we want to study. How­ever, when deal­ing with first or­der logic there will be many oc­ca­sions in which this is sim­ply not pos­si­ble, no mat­ter how many ax­ioms we add to our the­ory.

One re­sult show­ing this is the Skolem-Löwen­heim the­o­rem.

Ax­iom schematas

For proper de­duc­tion to work as in­tended, the set of ax­ioms of a the­ory do not have to be strictly finite, but just com­putablenoteIn­ci­den­tally, a the­ory whose set of ax­ioms is com­putable is called ax­iom­a­ti­z­able.

In par­tic­u­lar, we can spec­ify an in­finite amount of ax­ioms in one go by spec­i­fy­ing an ax­iom schemata, or par­tic­u­lar form of a sen­tence which will be an ax­ioms.

For ex­am­ple, in Peano Arith­metic you spec­ify the in­duc­tion ax­iom schemata, stat­ing that ev­ery sen­tence of the form \([P(0) \wedge \forall n. P(n)\rightarrow P(n+1)]\rightarrow \forall n. P(n)\) is an ax­iom of \(PA\).

The rea­son why first or­der logic can han­dle in­finite sets of ax­ioms is due to its com­pact­ness, which guaran­tees that ev­ery con­se­quence of the the­ory is a con­se­quence of a finite sub­set of the the­ory.

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  • Mathematics

    Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.