XPost: sci.logic, sci.math   
   From: janburse@fastmail.fm   
      
   Hi,   
      
   What you wrongly called Intermedite Value,   
   where Value refers to the Value of a function,   
      
   is only "dense order". Nobody calls it "gapless".   
   Because in the case of Q it has still gaps.   
      
   for x < y, there is z with x < z < y   
   https://en.wikipedia.org/wiki/Dense_order   
      
   For example Q has the irrational numbers as gaps.   
      
   Bye   
      
   Mild Shock schrieb:   
   > Hi,   
   >   
   > Take a = 1 and b = 2 with this function:   
   >   
   > f(x) = x^2 - 2.   
   >   
   > https://en.wikipedia.org/wiki/Intermediate_value_theorem   
   >   
   > Now take s = 0, what is x ?   
   >   
   > Mild Shock schrieb:   
   >> Hi,   
   >>   
   >> Looks like Kimi was grilling you with no   
   >> mercy. I guess you got a complete roast.   
   >>   
   >> BTW: You got Intermediate Value Theorem   
   >> theorem wrong. Its not about a 'c' such   
   >> that 'a < c' and 'c < b'.   
   >>   
   >> The important thing is there is also   
   >> a function f involved. In particular a   
   >> continuous function.   
   >>   
   >> Have Fun!   
   >>   
   >> Bye   
   >>   
   >> Ross Finlayson schrieb:   
   >>> Gaplessness (intermediate-value) inside ZF   
   >>>   
   >>> “IVP_d” ≡ ∀a,b∈E_d ∀k∈d+1 (a ≺_d b → ∃c∈E_d   
   (c=part_d(k) ∧ a ≼_d c   
   >>> ≼_d b))   
   >>> ZF ⊢ ∀d≥1 IVP_d  (finite intermediate-value property)   
   >>>   
      
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