Consistent mathematical definitions of McTaggart's A- and B- series

Here are consistent definitions of McTaggart's A-series and B-series. [refs.?] This does not address the question of whether a mathematical representation of time is sufficient to capture everything we know about time.

Let T be a timeline: a set of lineraly-ordered elements (which represent moments of time, or the state of the universe at moments of time, it doesn't matter), and T is isomorphic to the real numberline. Let

*t*1*,*t*2*, ... be variables that range over all times*t*in T.Start with the B-series. Define a (transitive) two-place relation B = _B_, such that for each pair of distinct times

*t*1,*t*2, we have(1)

*t*1B*t*2 if and only if*t*1 happens before*t*2The A-series is more subtle. As McTaggart famously noted, an element

*t*1 of T cannot be past, present, and future, simultaneously.One may start by picking some one time

*t*1 in T, and declaring it*present*.*t*1 is*now*(?...)*.*Then all the times*t*2 before it (in the B-series) are*past*and all times*t*3 after it (in the B-series) are*future*. These assignments are a particular (homomorphic) map M*t*1:{past, present, future} to T.The twist is that, given that the moment or time that is

*present*is (the value of the variable)*t*1, we need more, since other moments need time to be*present*as well, when it's their turn, in the linear ordering.Perhaps the most natural thing to do is to construct a map like M

*t*1 for*each*time in T. For each time*t*in T we have the map M*t*:{past, present, future} to T such that*t*is*present*, times before it are*past*and times after it are*future*. Then the set*S*of all maps M*t*,*S*= {M*t*for all*t*in T}, is a consistent assignment of (many of) each of the A-states*(past,**present,*and*future)*to each time*t*in T. In this sense McTaggart was wrong: each time*t*is associated with*past*by some maps M*t*',*present*by some M*t*'', and*future*by the remaining M*t*'''. You have to pick which time*t*is*present*. Since there is no reason to pick one time over another (so far as the mathematical representation is concerned) we needed the set of all homomorphic maps*(past,**present,*and*future)*to T.Conversely, one can start with the A-series and derive the B-series (or B-relation. I take a relation to be defined ultimately in terms of sets). Basically, if variable

*t*4's future is a subset of variable*t*5's future, then*t*4 is*after**t*5, i.e.*t*5B*t*4. The point is the B-series (relation) can be defined in terms of the A-series. Therefore(2) The mathematical structure of the A-series and the mathematical structure of the B-series can be defined in terms of each other.

The question is whether a mathematical representation of time is sufficient to capture everything we know about time.

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