How many different paradoxes are there

Confused and baffled the cabby gives the woman a discount, the cabby drives around the block for twenty minutes never realizing once there was no one in the car. But his pay check just doubled and the his meter reads as never having left the spot. Please consider these. You mentioned that the things in the bootstrap and predestination paradoxes occur in a cyclic and never ending fashion and you pointed it as the main problem.

If you look up the above sentence carefully you will notice that you the creator have tried to connect all these events in a cyclic fashion that you never wanted in the bootstrap paradoxes. Next you mentioned grandfather paradox as a paradox because the cycle is not completed. Both paradoxes are contradictory ………. How about the observers? Beings from the future that has traveled to our past and present. In their timeline they have no record of our life so they travel to the past to create a record for their own timeline.

Apparently there is some form of catastrophic event that happens and completely destroys all records prior. Are these observers changing our timeline by coming here, or does it not matter due to the fact the upcoming catastrophic event destroys all evidence of their presence in our timeline anyway.

This is why many of us think science is full of shit besides the fact it keeps changing. Why all this discussion about something that is obviously impossible? Only scientists …. I believe it would be close to The Multiverse hypothesis in that by traveling back in time, you end up in a parallell universe.

However, this parallell universe is not created neither when you travel or cause a time altering event. Imagine the amount of energy it would take to create it? To return to your own time line, I can only see two possibilities: a Time is not linear and everything that ever has, is, and will, all exist at same time. To then go into the future, you could end up in an infinite number of alternate realities. And that timeline would most likely not be the same you came from. Finally, would you be able to interact with yourself in a past timeline?

Either you died, or you were never born. Your knowledge of the future events would most likely also become false. Perhaps you know the winning lottery number and place all your money on it, only to lose it all.

Is it possible that you travel back in time and you will not see the same events which happened in your life. Here the question arise how you will confirm you are in same time line. My other concern is can we invent a device which can communicate across time dimensions just cellphone which connects across spaces. My other thought is quantum entanglement can happen across time.

Really liked how you broke down the theories for the regular person. Dina Rae, author. So try this, add quantum mechanics: by simply creating these concepts we are changing the universe, because it becomes what we want it to be.

So we create time travel and paradoxes because we all participate in stories that makes them real in our heads via reading, TV, etc. Like Star Trek, we all dreamed of communicators and tricorders, which in reality we now have. How long before they build a stargate? What do you call that paradox? Steins gate is somewhat true from what I have an understanding on but its not with tech. But viewing the what if situations that is possible our minds can dream up different paths we can take.

You might be able to affectedly take over an alternate version of yourself with enough will to do it. But that version of you must want it as well. When it comes to physically traveling to the past or future there will be A allot of power needed. B a map drawn up of all the possible situations that could happen to prevent the changes from reverting back and also the prevention of your own destruction of you meddling with things you should not be messing with.

Time travel would be super messy only in situations that its life or death should it be attempted. Knowing the future is just fooling ones self as if you trust what you see then it will happen but if you did not know what would happen anything is possible. If the person meets someone from the future say a relative or someone that is relative to the time travelers timeline will they be forced to see the endless time loop?

If so what if someone use powder to erase the effect will that work? Of course the same game also has changing things in the past changing the present, and a process in the past visible at the same speed in the present. Option 2: Quantum Mechanics might actually explain it. In the case of your grandfather, many actions might affect the future, but none of them could affect your ability to go back in time and do those actions, as the quantum wavefunctions necessary for this will cancel themselves out.

The ball making the original deviate and not go through the portal, but itself going through the portal again. Making the 2nd ball of unsure origin. I really believe that if Time Travel existed, The Bootstrap Paradox theory and Multiple universes hypothesis could be a better explanation, One would go back in time, Kill their grandfather, come back and find that they do not exist in that time line. However, the individual would have to share their fortune with their duplicated paradox Ref Continuum unless that individual kills their paradox self, they can live life to the full, unless the future paradox double goes back in time and creates a whole new time line while the previous paradox stays in that time line living their life, the new time line that person created which is a whole parallel universe idea, that is if the parallel universe infinite.

And such creation we call instantaneous. I posit the case that you see your father coming from a distance, in such a way that you do not discern whether it is your father or another.

Then it is proved, because you do indeed know your father, and he is the one approaching; hence, you know the one approaching. Likewise, you know him who is known by you, but the one approaching is known by you; hence, you know the one approaching. I prove the minor, because your father is known by you and your father is the one approaching; hence, the one approaching is known by you. Indeed, quite generally, sophisms about the nature of change and continuity, about knowledge and its objects, and the ones about the notion of self-reference, amongst many others, have attracted a great deal of very professional attention, once their significance was realised, with techniques of analysis drawn from developments in formal logic and linguistic studies being added to the careful and clear expression, and modes of argument found in the best writers before.

The pace of change started to quicken in the later nineteenth century, but the one earlier thinker who will also be mentioned here is Bishop Berkeley, who was active in the early eighteenth. For a history of this period, in connection with the issues which concerned Berkeley, see, for instance, Grattan-Guinness It will be remembered that in the calculation of a derivative the following fraction is considered:.

The point was appreciated to some extent elsewhere. Leibniz, however, had no problem with the notion of an actual infinite division of a line — or with the idea that the result could be a finite quantity.

However, while Leibniz introduced finite infinitesimals instead of fluxions, this idea was also questioned as not sufficiently rigorous, and both ideas lost ground to definitions of derivatives in terms of limits, by Cauchy and Weierstrass in the nineteenth century.

We will look at those in the next section, which will then lead us into twentieth century developments in the area of self-reference. These issues, it will be remembered, centred on the problem of non-recognition, and in various ways two central cases of this have been given close attention since the end of the nineteenth century. A great deal of other relevant discussion has also gone on, but these two cases are perhaps the most important, historically see, for example, Linsky Ortcutt, a respectable man with grey hair, once seen at the beach.

In one location he was taken to be not a spy, in another place he was taken to be a spy, as one might say; but is that quite the best way the situation should be described?

Maybe one who does not recognise him can have beliefs about the man at the beach without thereby having those beliefs about the respectable man with grey hair — or even Bernard J. Certainly Quine thought so, which has not only caused a large scale controversy in itself; it has also led to, or been part of much broader discussions about identity in similar, but non-personal, intensional notions, like modality.

Thus, as Quine pointed out, it would not seem to be necessary that the number of the planets is greater than 4, although it is necessary that 9 is greater than 4, and 9 is the number of the planets. A branch of formal logic, Intensional Logic, has been developed to enable a more precise analysis of these kinds of issue.

It was developments in other parts of mathematics which were integral to the discovery of the next logical paradoxes to be considered. These were developments in the theory of real numbers, as was mentioned before, but also in Set Theory, and Arithmetic. The tradition up to the middle of the nineteenth century did not look at these matters in this kind of way. For the natural numbers arise in connection with counting, for instance counting the cows in a field.

If there are a number of cows in the field then there is a set of them: sets are collections of such individuals. But continua from Cantor onwards have been seen as composed of non-finite individuals.

And not only that is the change. For also the number of individuals in some set of individuals — whether cows, or the non-finite elements in beef — has been taken to be possibly non-finite, with a whole containing those individuals being then still available: the infinite set of them. It is important to appreciate the grip that these new ideas had on the late nineteenth century generation of mathematicians and logicians, since it came to seem, as a result of these sorts of changes, that everything in mathematics was going to be explainable in terms of sets: Set Theory looked like it would become the entire foundation for mathematics.

In the latter case those things which are not members of themselves form a proper class but not a set, and proper classes cannot be members of anything.

The sets which are ordinals are so ordered that each one is a member of all the following ones, and so, with no limit envisaged to the sets which could be formed, it seemed possible to prove that any succession of such ordinals would themselves be members of a further ordinal — which would have to be distinct from each of them.

The trouble came in considering the totality of all ordinals, since that would mean that there would have to be a further distinct ordinal not in this totality, and yet it was supposed to be the totality of all ordinals. Cantor extended this theorem to his infinite sets as well — although there was at least one such set he realised it obviously could not apply to, namely the set of everything, sometimes called the universal set.

For the set of its subsets clearly could not have a greater number than the number of things in the universal set itself, since that contained everything. In fact an earlier paradox about the natural numbers had suggested that even they could not be consistently numbered: for they could be put into 1 to 1 correlation with the even numbers, for one thing, and yet there were surely more of them, since they included the odd numbers as well. This paradox Cantor took to be avoided by his definition of the power of a set N.

Thus all infinite sequences of natural numbers have the same power, aleph zero. But the number of points in a line was not aleph zero, it was two to the aleph zero, and Cantor produced several proofs that these were not the same. The most famous was his diagonal argument which seems to show that there must be orders of infinity, and specifically that the non-denumerably infinite is distinct from the denumerably infinite.

For belief in real numbers is equivalent to belief in certain infinite sets: real numbers are commonly understood simply in terms of possibly-non-terminating decimals, but this definition can be derived from the more theoretical ones Suppes , p But can the decimals between, say, 0 and 1 be listed?

Listing them would make them countable in the special sense of this which has been adopted, which amongst other things does not require there to be a last item counted. The natural numbers are countable in this sense, as before, and any list, it seems, can be indexed by the ordinal numbers.

Suppose, however, that we had a list in which the n-th member was of this form:. This seems to show that the totality of decimals in any continuous interval cannot be listed, which implies that there are at least two separate orders of infinity. Of course, if there were no infinite sets then there would be no infinite numbers, countable or uncountable, and so an Aristotelian would not accept the result of this proof as a fact. Discrete things might, at the most, be potentially denumerable, for him.

But Set Theory is one such theory, and in it, supposedly, there must be non-countable sets. Commonly it is accommodated by saying that, within the denumerable model of Set Theory, non-denumerability is represented merely by the absence of a function which can do the indexing of a set, that is produce a correlation between the set and the ordinal numbers.

But if that is the case, then maybe the difficulty of listing the real numbers in an interval is comparable. Certainly given a list of real numbers with a functional way of indexing them, then diagonalisation enables us to construct another real number. But maybe there still might be a denumerable number of all the real numbers in an interval without any possibility of finding a function which lists them, in which case we would have no diagonal means of producing another.

We seem to need a further proof that being denumerable in size means being listable by means of a function. Consider for a start all finite sequences of the twenty six letters of the English alphabet, the ten digits, a comma, a full stop, a dash and a blank space. Order these expressions according, first, to the number of symbols, and then lexicographically within each such set. We then have a way of identifying the n-th member of this collection. Now some of these expressions are English phrases, and some of those phrases will define real numbers.

Let E be the sub-collection which does this, and suppose we can again identify the n-th place in this, for each natural number n. One significant fact about this paradox is that it is a semantical paradox, since it is concerned not just with the ordered collection of expressions which is a syntactic matter , but also their meaning, that is whether they refer to real numbers.

It is this which possibly makes it unclear whether there is a specifiable list of expressions of the required kind, since while the total list of expressions can certainly be straightforwardly ordered, whether some expression defines a real number is maybe not such a clear cut matter. Indeed, it might be concluded, just from the very fact that a paradox ensues, as above, that whether some English phrase defines a real number is not always entirely settleable.

So is there a definite set of English expressions which name integers not nameable in less than nineteen syllables? Mackie disagreed with Ramsey to a certain extent, although he was prepared to say Mackie , p :.

The semantical paradoxes…can thus be solved in a philosophical sense by demonstrating the lack of content of the key items, the fact that various questions and sentences, construed in the intended way, raise no substantial issue. Specifically he held that statements about all the members of certain collections were nonsense compare Haack , p :. Whatever involves all of a collection must not be one of a collection, or, conversely, if, provided a certain collection had a total it would have members only definable in terms of that total, then the said collection has no total.

But this, seemingly, would rule out specifying, for instance, a man as the one with the highest batting average in his team, since he is then defined in terms of a total of which he is a member.

This approach, however, presumes that semantical notions, like definability, designation, truth, and knowledge can be construed in terms of mathematical sets, which seems to be really the very supposition which Ramsey disputed. Does this contradiction mean there is no such concept as heterologicality, just as there is no such set as the Russell set? The point is made even more plausible given the very detailed logical analysis which Copi provided Copi , p And Copi gives no proof of this.

The Liar Paradox is a further self-referential, semantical paradox, perhaps the major one to come down from antiquity. And one may very well ask, with respect to. But there is a well known further paradox which seems to block this dismissal. If I am saying nothing meaningful here, then seemingly what I say is true, which seems to imply that it does have meaning, after all.

Let us, therefore, look at some other notable ways of trying to escape even the Unstrengthened Liar. The Unstrengthened Liar comes in a whole host of variations, for instance:. Even Socrates is baffled. It's a crucial insight from one of the founders of Western philosophy: You should question everything you think you know.

Indeed, the closer you look, the more you'll start to recognize paradoxes all around you. Read on to see our favorite Catchs from Wikipedia's epic list of more than types of paradoxes. The dichotomy paradox has been attributed to ancient Greek philosopher Zeno, and it was supposedly created as a proof that the universe is singular and that change, including motion, is impossible as posited by Zeno's teacher, Parmenides.

From a mathematical perspective, the solution — formalized in the 19th century — is to accept that one-half plus one-quarter plus one-eighth plus one-sixteenth and so on This is similar to saying that 0. But this theoretical solution doesn't actually answer how an object can reach its destination. The solution to that question is more complex and still murky , relying on 20th-century theories about matter, time, and space not being infinitely divisible.

This is called the arrow paradox , and it's another of Zeno's arguments against motion. The issue here is that in a single instant of time, zero seconds pass, and so zero motion happens.

Zeno argued that if time were made up of instants, the fact that motion doesn't happen in any particular instant would mean motion doesn't happen. As with the dichotomy paradox, the arrow paradox actually hints at modern understandings of quantum mechanics. In his book " Reflections on Relativity ," Kevin Brown notes that in the context of special relativity, an object in motion is different from an object at rest.

Relativity requires that objects moving at different speeds will appear different to outside observers and will themselves have different perceptions of the world around them.

Another classic from ancient Greece, the Ship of Theseus paradox gets at the contradictions of identity. It was famously described by Plutarch:. The ship wherein Theseus and the youth of Athens returned from Crete had thirty oars, and was preserved by the Athenians down even to the time of Demetrius Phalereus, for they took away the old planks as they decayed, putting in new and stronger timber in their places, in so much that this ship became a standing example among the philosophers, for the logical question of things that grow; one side holding that the ship remained the same, and the other contending that it was not the same.

While we're at it, how can evil exist if God is omnipotent? And how can free will exist if God is omniscient? These are a few of the many paradoxes that exist when you try to apply logic to definitions of God.

Some people might cite these paradoxes as reasons not to believe in a supreme being; however, others would say they are inconsequential or invalid.

Moving ahead to a problem posed in the 17th century, we've got one of many paradoxes related to infinity and geometry.

As stated in the MathWorld article on the horn , this means that the horn could hold a finite volume of paint but would require an infinite amount of paint to cover its entire surface. Here is one of many self-referential paradoxes that kept modern mathematicians and logicians up at night.

An example of a heterological word is "verb," which is not a verb as opposed to "noun," which is itself a noun. Another example is "long," which is not a long word as opposed to "short," which is a short word.

So is "heterological" a heterological word? If it were a word that didn't describe itself, then it would describe itself; but if it did describe itself, then it would not be a word that described itself.

This is related to Russell's Paradox , which asked if the set of things that don't contain themselves contained itself. By creating self-destructing sets like these, Bertrand Russell and others showed the importance of establishing careful rules when creating sets, which would lay the groundwork for 20th-century mathematics.

Protagonist Yossarian is introduced to the paradox with regard to pilot evaluation but eventually sees paradoxical and oppressive rules everywhere he looks.