Free Will is incompatible with Foreknowledge

Definitions:q is an atomic proposition[1]s_i is a set of consistent[2] atomic propositions known as a "state"States represent "possible worlds"[3]. Any collection of consistent (non-contradictory) propositions could possibly be true in a logical sense, thus any such state represents a world that could be or could have been.Not all such worlds are "accessible" in the sense that there exists a series of events that could lead to such worlds coming into being. Many "possible worlds" could have been given a different past but will never come to be given the actual present. To represent accessible worlds (possible futures) we use the following definition:(s_i, s_j) is an ordered pair of states known as a transitionTransitions represent progression through time. The second state in the pair represents a possible future with respect to the first state. In other words, if the first state is the actual present, then the second state represents a possible future that could come to be.C(a,b) - a free will choice between mutually exclusive options a and bF(x) - x is foreknown The premises and the argument will be based primarily in predicate logic[4]. While I will provide prose interpretations of the premises and the argument, the formal expressions are what are in contention here. The formal expressions were created using LaTeX and uploaded as gifs. Explanations will follow each section.(For some reason, it is not letting my upload pictures, please use the links)https://www.debate.org...P1. If there is a free will choice between (s_i), then:there exists a state (s_j) where a is true and that state is a possible future to s_i;and there exists a state (s_k) where b is true and that state is a possible future to s_i.P2. Formal representation of the mutual exclusivity of a and b. If b is true in a state, then a is false (and vice versa).Argument, part 1:https://www.debate.org...1. We will suppose that there is a free will choice between a and b in state s_0.2. Therefore, according to P1, there exists a state (s_j) where a is true that is a possible future to s_0; and a state (s_k) where b is true that is a possible future to s_0, using modus ponens[5].3. This is simple conjunction elimination from #2[6].4. #3 says there exists such a state s_j; we will represent this state as s1a using existential instantiation[7].5. Therefore, "a is true" is foreknown in s_0, given 4 and P3, using modus ponens.Summary: We have established foreknowledge of a being true in s_0.Argument, part 2:https://www.debate.org...6. This is simple conjunction elimination from #2.7. #6 says there exists such a state s_k; we will represent this state as s1b using existential instantiation.8. This is simple conjunction elimination from #7.9. Since b is true in s1b, then a is false, given P2.10. This is simple conjunction elimination from $7.11. We conjoin #9 and #10 using conjunction introduction[8].12. Therefore, "a is false" is foreknown in s_0, given 11 and P3, using modus ponens.Summary: We have established foreknowledge of a being false in s_0.Argument, part 3:https://www.debate.org...13. We conjoin #5 (from part 1) and #12 (from part 2).14. Since each is individually true in s_0, then the conjunction is true in s_0, given P4.Summary: We have both foreknowledge that a is true and foreknowledge that a is false in the same state (s_0)Conclusion: This contradicts the given premise P5, where such foreknowledge is not possible. This either disproves either: our supposition in #1, or one of the premises via reduction ad absurdum[9].I contend that P1, P2, P4 and P5 cannot be thrown out, as they are so fundamental as to be axiomatic.This means we have to dismiss our supposition (that a free will choice exists) or P3 (that we can foreknow the outcome of such a choice), proving that Free Will is incompatible with Forewknowledge.Con Must:Demonstrate some invalid inference in the argument;or argue for different premises that remove the contradiction and explain why Con's premises are superior to Pro's.[1] http://en.wikipedia.org...[2] http://en.wikipedia.org...[3] http://en.wikipedia.org...[4] http://en.wikipedia.org...[5] http://en.wikipedia.org...[6] http://en.wikipedia.org...[7] http://en.wikipedia.org...[8] http://en.wikipedia.org...[9] http://en.wikipedia.org...