How to do paradox free time travel

TL;DR: copy on write

First of all there's nothing wrong with your story having a time paradox, in fact certain stories require it. Trying to prevent a paradox is a valid motivation for a crisis, timestream changes can be an interesting way to add complications, etc. However for some stories you may want it free of paradox and that's more tricky than it sounds so I thought I would detail how this can be done (option #3 is the only real way).

First I need to explain what the issue is. For a single timeline that can be altered by traveling backwards in time there are 2 paradoxes that arise: a grandfather paradox and a bootstrap paradox. The grandfather paradox is the most commonly talked about paradox (and is normally just called a "time paradox" but that would be confusing). A grandfather paradox is often described as "what happens if I go back in time and kill my grandfather before my father was conceived?" The problem is that if there is a single timeline (by now you may have guessed the solution) on which you go backwards you can contradict it. If you kill your grandfather then you couldn't've been born which means you couldn't've killed him which contradicts the original action and thus the timeline is in an invalid state.

A bootstrap paradox is sort of the opposite. An example would be that you have a necklace that has been passed down from your great grandfather. You hear a story that he got it from a time traveler who turns out to be you. In an attempt to avoid a grandfather paradox you go back in time and give the necklace to your great grandfather. The necklace's place on the timeline is a nice circle which appears to be fixed but who created the necklace? How old is it? This necklace exists because it already exists. This is a causation paradox (which requires backwards time travel).

Notice that if you go back in time and do something then it will either cause a grandfather paradox (because the event wasn't supposed to happen) or a bootstrap paradox (because it was supposed to happen). You may say "I promise not to do anything important" but that's not good enough. Even something as simple as stepping on an ant is an event that happened and thus would cause either paradox but it's more extreme than that: every breath you take, every photon of light you absorb, every air molecule you displace. It's impossible to exist in any tangible way without causing a paradox.

However there are solutions which I will do from least useful to most. The first point (option #0) is to not do time travel at all which works but doesn't meet the criteria of "paradox free time travel" and thus is not considered an option. The first actual option (which isn't much better) is to do forward only time travel. These paradoxes only occur from going backwards, going forward causes no issues as long as the traveler never returns (by going backwards). You could make a story of someone going into the far future and be unable to go back and there would be no paradox (he absolutely can't go back to "his time" since that would cause a paradox with the present he's currently in). In fact this has been demonstrated to be possible in real life (only going forward a few seconds) and thus forward only time travel is an undebatable scientific occurrence that's iron clad against any paradox.

Option #2 is to have the past be "read only" such that it can be seen but nothing can be changed. The man who goes back will be as though he is intangible and absolutely nothing he does makes any difference. For this to work the time travel device would need to duplicate the photons so he can see them (in addition to the past photons which the past objects interacted with) and the same is true for all his senses. You may wonder if this is actually time travel or just watching a movie of something that happened in the past. You can describe it as backwards time travel but it doesn't really feel like it is.

This leaves the only real option #3 which is complicated but can be summed up as "copy on write". The 2 paradoxes hinge on the phrase "a single timeline that can be altered by traveling backwards in time." Option #1 disallowed traveling backwards, option #2 disallowed altering, and option #3 doesn't use a single timeline. Whenever someone travels back in time by any amount the timeline is copied and they are placed in the new one. Traveling forward in time remains in the same timeline. Basically the past of each timeline is unchangeable but a copy can be changed in any way without paradox. This model allows meeting copies of yourself since if you travel back in time to a point you existed the entire past is copied including your body. If you don't like what you've changed you can go back and murder your copy who was about to change it (this doesn't affect the previous timeline you were in) or go back right before he got there and have no copy show up (since only the past is copied). You can get as many copies of yourself as you like and if you'd like to return to normal you can travel back to before you touched anything then forward in that copy to your normal time to get an equivalent to your original assuming your brief tangibility didn't affect much. You could also permit returning to a timeline as long as the point of return is during or after what is considered the present for that timeline (the present being the last point in that timeline that has been visited by anyone).

Option #3 sounds complicated but allows for basically everything and should be easy to use. In fact if you never explain that copies are being made this time travel model will "just work" and still allows things like "we need to stop this event or we are screwed" for the timeline you are in. That said this model (as is) doesn't allow someone to be aware that the timeline has changed (since it's a copy, their own timeline hasn't changed) and again a crisis like "if we don't prevent this time paradox it will be the end of time itself" would require a paradox based model.

Note that being able to predict the future causes no time paradox. It doesn't matter if the prediction is contradicted. Unlike traveling to the future and back which causes issues because it wasn't a prediction, it actually happened and things aren't supposed to un-happen.

Professor Farnsworth: Your grandfather?! Stay away from him, you dim-witted monkey! You mustn't interfere with the past! Don't do anything that affects anything, unless it turns out you were supposed to do it, in which case for the love of God, don't not do it!

Fry: Got it.

Professor Farnsworth: If, for example, you were to kill your grandfather, you would cease to exist.

Fry: But existing is basically all I do!

--Futurama episode "Roswell that Ends Well" (TV 4x1). A joke that only works if you allow paradoxes.

Arbitrary guesswork is the foundation of all knowledge

I haven't read "Gödel, Escher, Bach" but based on its description it sounds like it beat me to this idea. That said I'm going to explain this in a single post (much shorter than a book) and making it as easy to understand as I can. That said it's probably still going to be pretty thick.

I'll start by paraphrasing Gödel's first incompleteness theorem: "All systems must have at least 1 axiom." An axiom is something that is accepted to be true without being proven. Axioms are core basic beliefs that everyone has many of such as "I exist," "other people are not my imagination and they exist in the same way I do," and "I have senses which let me measure the world I live in." Axioms can't be proven since you have to use something as a basis for your argument. Using an axiom to prove itself is circular reasoning. Although it is possible for an axiom to agree or disagree with another axiom there must be at least 1 that is assumed to be true on which all other claims are based.

"How do I know what axioms to trust? You can't prove to me that I exist!" The fact that you can read this article shows that you already have chosen some axioms. While there are some basic axioms that everyone needs to survive how are the rest chosen? "I'm going to choose axioms that meet this criteria." That would make that criteria the axiom on which the others are based. Since axioms are so fundamental there's no getting around the fact that they are indeed "accepted without being proven" or even recommended. This is to say that all axioms are chosen arbitrarily, at random, and without reason. "I like axiom A more than axiom B" that's an example of being chosen arbitrarily. "Axiom A is consistent with my other axioms" then those other axioms and the desire for consistency are you base ones. It's fine to change axioms or to have an axiom demoted to being based on others.

Have I lost you yet? I'll explain it from a different angle. Let's suppose you have some bit of knowledge that isn't an axiom such as "you need to stir the noodles so they don't stick to the pan." Consider what that knowledge is based on. And what is that based on. And what is that based on etc. Eventually you can follow it back to at least 1 bit of knowledge which is not based on any other knowledge. That most fundamental knowledge is an axiom. This is the "first mover" argument except with knowledge it is impossible to claim that your knowledge is an infinite series backwards. Unless you believe that you have infinite knowledge (in which case why are you reading this?) then you must believe that going backwards will eventually reach an end (an axiom or a group of them).

"Of course I trust that my senses can detect the world. There's no other way to live." But how much do you trust them on a scale of 0 to 100%? You can't trust them 0% because if you don't trust pain at all then there's no reason for you to eat or breath which leads to a quick death. However you also shouldn't trust them 100% of the time because senses can be tricked by slight of hand or hallucinogens. 100% trust is much less dangerous than 0% but it can still lead to trouble (people tricking you or a drug addiction). So you need to trust your senses only sometimes. How do you know when to trust them? It's based on something else you trust. "Of course a person can't just disappear, it's some kind of stage magic" is based on axioms. But obviously you can't use your senses to decide which knowledge should be chosen as a basis for when to trust your senses.

Consider also how much you trust other people. It can't be 0% if you plan to live around people and interact: "if I work for you how do I know that you'll pay me?", "if I pay you how do I know you'll let me walk out with these groceries?", and "if I get near you how do I know you won't try to kill me?". But it shouldn't be 100% either (evil people exist). "Who you should trust" is a more interesting question than "when to trust." Growing up most people trust their parents by default who in turn instill many ideas. Later in life those ideas can be questioned but the ideas we grow up with are likely foundational. Therefore what we grow up with (whether it matches our parents or not) may stick our entire life since they were the original axioms (assuming they survived teenager years).

"I could try to find axioms that don't contradict." First of all your desire for no contradiction and the criteria you use to judge that are based on axioms. Secondly there are plenty of axioms that don't contradict so you'll need to chose from among them arbitrarily. Thirdly in the case of contradiction you'll need to decide which one to keep arbitrarily.

Have I driven home the point that all knowledge is arbitrarily based yet? This argument is a logical consequence of Gödel's first incompleteness theorem. Of course this article depends upon axioms so you may disagree given different axioms (or even the same ones). But since you must have at least 1 axiom the point remains: that axiom was chosen without reason.

I'd like to end by giving this article a practical application (as left field as it may be). Given that all knowledge is arbitrarily based there's room to be compassionate for even the craziest or most offensive ideas/opinions. If you hear someone who is for or against some hotly debated topic (eg abortion, homosexuality) know that they just have very different axioms than you and the way they chose those axioms was just as valid as how you chose your's. Therefore even in strong disagreement you can have a calm loving conversation. If someone think aliens, witchcraft, and ghosts are a good explanation/reason don't call them crazy or criticize because from their point of view you seem just as crazy.