The first case: 2 yuan 1 bottle of wine, 4 bottle caps are 1 bottle, and 2 bottles are 1 bottle. The calculation is as follows: 1 bottle cap 0.5 yuan, 1 bottle/yuan, 1 bottle.

In the second case, 10 yuan bought 5 bottles of beer and borrowed 15 bottles of wine, so Party A drank 20 bottles of beer. Replace 20 empty bottles with 10 beer, 20 bottle caps with 5 beer, and 1 bottle with 15 beer. * * * Number of drinking bottles =5+ 15=20 bottles of wine.

"Mathematical Problems" is an anthology of Hilbert's speech "Mathematical Problems" at the International Congress of Mathematicians in Paris from 65438 to 0900. The 23 mathematical problems he raised in his speech stimulated the imagination of the whole mathematical field and promoted the development of mathematics in the 20th century. In this speech, Hilbert also expounded his incisive views on the essence of mathematics, the source of mathematical knowledge, the importance of mathematical problems and research methods.

1963 paul cohen, an American mathematician, proved by coercion that ZFC could not deduce the continuum hypothesis. In other words, whether the continuum hypothesis is established cannot be decided by ZFC.