If I understood it right, you defined an equivalence on a time period (let's say 10 years). What I mean, is that in a space of states of 10 years (example) of possible transactions, systems can be equivalent for 5 minutes, 10 minutes, 3 years or whatever. They are equivalent on this time frame, but not on the whole space of states. Another point would be your reference to the infinity of time. You just refer to it as an increasing time period, but you forget that your time scale can go subsecond and create an infinity at that level. This is why mathematicians tend to consider their models on a continuous time frame, in order to represent observations of everyone.
Then your conclusion is wrong if you accept that trades can be put in continuous time (another philosphical debate...)
Thanks for not resorting to name calling. I'll need to chew on it first (though my conclusion might not necessarily be same as yours kekeke) as I have yet to learn calculus beyond the very basics. Just to clarity : Your belief is that for any set of data, say ES data for '95-'00, an infinite number of systems exist that could work on it?
You know the saying 'more than one way to skin a cat'? Well for every different 'way' thats found you add one. You dont add zerro on the basis that the result is the same. Skinning a cat is a good example of trading in that its a verry simple rssult we want, make money, buy low sell high. But the number of ways you can go about it and the amount of extra information you can use is phenominal (unlimited even). I know traders who have made systems based on moon phases.
It doesn't matter that two systems might 'track together for some time, if they're different then they will eventually show different trades and need to be considered as different. As long as we are prepared to consider the example of real big numbers as finite, then we need to consider the systems in a market going forward way more than ten years. We need to consider tousands of years or more, when slight differences in systems will surely show up.
it doesn't matter if there are are infinite or finite number of systems to be created, since if there is a finite number none of us will ever come close to exhausting the potential if you feel limited to what you perceive as the 'finite' systems your not a very good systems developer start from scratch and try to make a simple MA crossover indicator into a working system, because it definitely can be done
After these many comments I realised that I committed two fallacies at the beginning of this thread: 1) That mechanical system developers are bound to using the entire history of a market during backtest. 2) That there aren't systems that last forever, or so long that it doesn't matter (10yrs?). 3) That the number of finite profitable systems that exist is not so large that can be consider infinite. I'm intrigued by this comment. Perhaps it'll make a good topic for another thread. ;-)
What indicator could you use for an automated system that has several lines and one of them tracks (gives a value for...) volatility? The higher the volatility the better your signal is for entry....or at least if the volatility is low at entry, then your volatility value should at least be increasing to give the best entry signal. Trade systems that use a volatility value/measurement as one of your entry conditions are the trades that will give you very powerful profits....trade entries into high or increasing volatility work the best IMO. Automated trade entries in a low volatility environment or one that is decreasing give you most of those "chop" type trades that whittle down your equity curve....as I have said before in this thread, find a way to include a value/measurement of volatility to add as one of your conditions for very strong system yield curves. Merry Christmas All!!!
Okay, you asked the direct question. My direct answer, is, yes, I believe there are an infinite amount of ways ES could be traded profitably from 1995 to 2000. Of course there are not an infinite number being traded. But the math supports the belief. So basically that idea gives a whole lot more reference material to one who is looking for a new edge. How much more effective do you think one would be carrying around the idea that there is ALWAYS an edge, than one that believes that there are a limited amount of edges, and all the good ones are gone? But in honest, that probably edge (the system) is one part of the necessity. The other part is: hey....can you trade that system as you designed and tested it? It makes no difference if you plug into the infinite combinations available if you don't find one that you can actually trade given your personal makeup.