Hi guys.

If I may make this humble observation on the starting manipulation used in 'proofs' which has 10 x 0.999... = 9.999...?

I naively observe for your joint consideration that when we multiply by 10

**we effectively add a "0" to the last place of a string**. Yes?

BUT

**in the expression 0.999... there IS no 'last place' in that string,** so the multiplication by 10 cannot logically add a "0" at the last place in order to 'shift' the first 9 to the left' to in front of the 'decimal point' (in that decimal notation format).

Else we would have 9.999...0. Which would be a nonsense in the same axiomatic treatment which multiplied by ten. Yes?

This illustrates what QQ has been pointing out all along. Ie, unless there is an 'infinitesimal' added on the end of the 0.999... string, we cannot go from there to here by adding a "0" via multiplication by 10.

My further humble observation is that QQ seems to be the only other person here that recognizes the importance of "contextual axioms" rather than isolated axioms which when followed blindly lead up a dark alley where trivial and conta/undefined situations occur which must require further isolated 'exceptions axioms/definitions' to paper over the axiomatic gaps which open up as QQ and others have tried to point to.

That's all I ant to observe at this juncture. I only did so at this time because I felt that what QQ was trying to illustrate was being unintentionally 'hidden' by 'proofs' from others which depended on (to me) trivial manipulations of algebraic symbols and algebraic 'rules' which make no sense when brought back to the fundamental actions. One example of which is the multiplication by 10 of the string 0.999... and assuming (without proper care or cause) that the result is 9.999... instead of 9.999...0 (which last place "0" would logically represent the 'infinitesimal' which QQ is referring to?).

In the universal reality there IS a "last infinitesimal of PHYSICAL effectiveness' which delineates the border/boundary 'condition between effective reality and ineffective reality scaleextent/strength etc of the various fundamental physical forces/entities which produce the Quantum Mechanics arena/phenomena we treat in reality via maths and logical modeling which finds its LIMITS in reality to that "last infinitesimal QUANTUM of physical effectiveness" which our quantum Mechanics already recognizes mathematically and logically in its modeling constructs. Yes?

That is why IN MY OPINION and obesrvation, contextual maths and reality physics are actually logically 'one' and consistent when treated under contextually complete 'rules' rather than partial 'isolated' axioms/postulates, because there IS a last infinitesimal of effectiveness in both any number string and in any physical modeling construct. In MATHS, that last infinitesimal represents the final 'quantum step' into a new STATE OF TRANSITION/BALANCE etc depending on what the process/states on either side of the continuum division involves in math/physical reality/properties.

Thanks for your exhaustive and interesting discussions/contributions, everyone! I have appreciated it all, every bit. Bye and good luck in all your discussions, QQ, Pete, everyone.