Abstract: Say a first-order theory (or a type) has bounded axiomatization if it has an axiomatization by ∀_n-formulas for some finite n. In this talk, we will discuss basic properties of theories and types with (or without) bounded axiomatizations, and in particular whether boundedness of theories implies that of types. (The meaning of the title will be explained in due time.)

Abstract: You have a candle that burns for a minute. How can you measure half a minute with it? Answer: Light both ends. If you light one end of another candle and light the other end in half a minute, you get 45 seconds. Now if you have an infinite supply of candles (unfortunately not provided), what are all the times that can be measured exactly? These are called the "fusible numbers." In this talk, we will explore what these numbers look like on the real line and, more importantly, when such facts are not provable.