Does false self = no self? My new understanding (based on this post) is that the self is not permanent and is always changing. However, I still can't make the logical assertion that the self does not exist at all. At this point, it makes more sense for me to say: "there are multiple versions of the self that pass through time, each of those selves being slightly different than the ones before. So, I am self version 1.0.1, self version 1.0.2, self version 1.0.3, etc".

In short, here are my present assertions: 


a configuration of energy and matter intersects with the 4th dimension of time over and over again, producing the by product is what we understand as self.
if some version of self did not exist, then the subject of knowledge could not be observed. Example: without a self, there would be no one to respond to this post.


Are my assertions correct? Does Buddhist philosophy teach that the self does not exist or merely that it is impermanent?
Imagine a river flowing over some rocks. At one point where the river bed makes a sudden turn, the flowing water forms eddies - little spinning whirlpools of water.

Does an Eddy exist? Well, yeah. Is it permanent? No. But it gets more interesting than that. 

Does an Eddy exist intrinsically, as an entity? -- now, this is a more interesting question...

The water an eddy is made from is different every second. And yet it looks like the same eddy stays in place for minutes or even hours.

Looking at one particular Eddy - what is it, really? What is its real identity? Is the water = Eddy? No! Is this bank = Eddy? No! Is that bank = Eddy? No! Is this section of the river bed = Eddy? No! But if you take exception to the water, to this bank, to the other bank, to the river bed - where can you find the Eddy?

It is exactly thanks to the water flowing, thanks to this bank, thanks to the other bank, thanks to the river bed - that here emerges a little whirlpool, an eddy.

In Buddhism we call this "samskara" (Pali "sankhara") - an assembled phenomenon or simply an assembly. Assemblies are not just impermanent - we say they are "empty of identity" or "coreless". Meaning, if you deconstruct any one of them, like I just did with this eddy - you will not find any core, any "main thing" that is responsible for its existence and for it staying itself. Assemblies emerge from interplay of multiple causes and conditions.

According to Buddhism (both Theravada and Mahayana) "self" as well as all other phenomena are assemblies. They are not just impermanent, they are empty of core, empty of identity. They are not entities, they are patterns of behavior emerging from interplay of causes and conditions.

Because self is an assembly, a different self can get assembled in different circumstances. For example, when you visit your parents, the self assembled with them is probably different from the self assembled at work, or the one assembled at the bachelor party. The feeling that self is the same entity that has continuity from moment to moment is a perceptual illusion, just like with Eddy. In fact, both our perceptions as well as our actions come from interplay of causes and conditions, with identity overlayed on top by the imputing conceptual mind (which is itself an interplay).

This is why when Buddha was asked whether Tathagata exists after death, he said, Tathagata cannot be identified even when alive, it's an assembled phenomenon emergent from an interplay of factors, including such factor as the presence of students imputing the label "Buddha" on the teaching situation.