Can you present the two propositions that are incompatible?
Yes I can if you'd like although im sure youve heard these arguments before. One the Bible says that the Earth was created in 6 literal days and rested on the 7th. Read Exodus 20:8-11 where Moses writes:
"Remember the Sabbath day by keeping it holy. Six days you shall labor and do all your work, but the seventh day is a Sabbath to the Lord your God. On it you shall not do any work, neither you, nor your son or daughter, nor your male or female servant, nor your animals, nor any foreigner residing in your towns.
For in six days the Lord made the heavens and the earth, the sea, and all that is in them, but he rested on the seventh day. Therefore the Lord blessed the Sabbath day and made it holy."
It wasn't just the book of Genesis that Moses said that God made the Heavens and the Earth in six literal days . These weren't generations of billions of years but in fact seven literal days. When we look at the Hebrew it further proves that this was what Moses's original intention to explain that these days were in fact literal. The Jews knew this and understood this for thousands of years.
How can God make the universe and everything in it in seven days if it was made over billions of years? Two, the Bible says that the universe existed on the 3rd day in Genesis 1: 14-19 Moses said that the sun, the stars, . If the big bang is true and the world started with a giant explosion on the first day of creation why did Moses say that it was on the 3rd day that the sun, moon and stars were created? Which would have been the 1st day of creation if the big bang is true?
3. Modern science still hasn't 100% explained these descrepencies or explained the origin of life and why we are here. Modern science is treated as fact when they are just in fact, theories as to the origin of life. Whereas the Bible flat out says how life began and why we are here. God created universe and everybody in it. Evolution provides a theory as to the origin of life and its merely just a theory. Until it can be replicated like the rest of science it's won't be any more than just a theory whereas the Bible can be trusted because it's been proven over and over again accurate. The Bible says God created the world in six days therefore that's how life started if people take the Bible seriously for what it says.
4. Carbon dating isn't really 100% accurate as to the age of fossils because scientists assume that the amount of carbon 14 in the atmosphere has been constant over the course of time the pre-flood environment could possibly have had different levels of Carbon 14 therefore affecting the accuracy of the results. We know that the fossils we have is after the flood because God killed off everything on the Earth during the flood including the dinosaurs and everything that lived.
There are more but these are the main arguments for Creationism. If you want to learn about more evidence you can read Answers in Genesis or CMI industries they both do fantastic work at proving Creationism.
Btw, I wasn't saying that Evolution is in fact, wrong there just isn't any 100% definitive evidence for it but we do have definitive evidence of our world's history in the Biblical accounts of Jewish history. According to Jewish history that has been passed down since the time of Moses the world is approximately 7,000-10,000 years old. I tend to believe the nation that started everything and the man who saw the very back of God than a theory that hasn't 100% been proven to be true when God has imo, been 100% proven to be true.